this is really impressive thank you. It its too advanced for me, so I have to study those functions and methodology of yours, nevertheless thank you, I appreciate it.

The short answer is that as this was a year ago I have no recollection of the problem, let alone why I saw fit to code it as I did. The next best thing I can do is show intermediate results from running a smaller example. This might make it easier to see what happens at each step.

It also might be helpful to look at what ListConvolve does. Here is a short example.

ListConvolve[{a, b, c, d}, {1, 2, 3, 4, 5, 6}]

(* Out[682]= {4 a + 3 b + 2 c + d, 5 a + 4 b + 3 c + 2 d, 
 6 a + 5 b + 4 c + 3 d} *)

In effect it lines up the first vector under the second one, multiplies elementwise and sums, then slides the first vector one place to the right, repeats, and continues in this way until they are aligned on their right ends (they start out aligned on the left ends).

Also have a look at documentation for Nest. It is a neat way to iterate a function application without forming an explicit looping construct.

Here is the code with some Print statements in place for showing intermediate results.

gSum[t_, d_] := 
 Module[{iter = Ceiling[101 - t], vec, row}, 
  row = ConstantArray[0, 2*d + 1 + 10*iter];
  row[[d + 5*iter + 1]] = 1;
  vec = Reverse@{pa1, 0, pa2, 0, pa3, p0, pb3, 0, pb2, 0, pb1};
  Print[row];
  Print[vec];
  Range[-d, d].Nest[(Print[#]; ListConvolve[vec, #]) &, row, 
    100 - t + 1]]

Now we try it out on a "small" example, where we decrease the number of iterations and also d.

gSum[90, 2]

(*During evaluation of In[677]:= {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}

During evaluation of In[677]:= {0.0187268926056,0,0.0112361355634,0,0.00216035798122,0.926427083333,0.00278755868545,0,0.0144982394366,0,0.0241637323944}

During evaluation of In[677]:= {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}

During evaluation of In[677]:= {0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.0187268926056,0.,0.0112361355634,0.,0.00216035798122,0.926427083333,0.00278755868545,0.,0.0144982394366,0.,0.0241637323944,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.}

During evaluation of In[677]:= {0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.000350696506663,0.,0.000420835807996,0.,0.000207164326207,0.0346982009931,0.000152952774553,0.0208189205958,0.000610323866969,0.004002828287,0.859510016593,0.00516493972516,0.000613427203787,0.0268631233495,0.000185234010799,0.0447718722491,0.000344914590979,0.,0.000700663155872,0.,0.000583885963227,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.}

During evaluation of In[677]:= {0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,6.56745581745*10^-6,0.,0.0000118214204714,0.,9.36574227431*10^-6,0.000974684225409,7.07879972564*10^-6,0.00116962107049,0.0000198532031615,0.000575767927495,0.0482636460291,0.000425098778452,0.0289711879004,0.00169626167989,0.00558446399082,0.798576225484,0.00720328490394,0.00170488672572,0.0373819616221,0.000514817413076,0.0622758072744,0.000958614655559,0.0000265440325772,0.00194733997168,0.000012691141759,0.00162278330973,0.0000201204224617,0.,0.0000253959554956,0.,0.0000141088641642,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.}

During evaluation of In[677]:= {0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,1.22988039786*10^-7,0.,2.95171295485*10^-7,0.,3.22406388303*10^-7,0.0000243370757515,2.81644340655*10^-7,0.0000438067363527,5.99729963139*10^-7,0.0000347067091938,0.00180739653767,0.0000262319671333,0.00216888988998,0.000073570180399,0.00106816189014,0.0597296922676,0.000789050728104,0.0358859906172,0.00314548877856,0.00695241079309,0.743026294279,0.00896168069601,0.00316154466301,0.046303702426,0.000955595604231,0.0770710708598,0.00177829166283,0.0000983644427217,0.00361104884579,0.0000470296697758,0.00300921960408,0.0000745604171864,1.09488243574*10^-6,0.0000941100039131,6.99648969278*10^-7,0.0000522833355073,8.93710434366*10^-7,0.,8.18214763327*10^-7,0.,3.40922818053*10^-7,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.}

During evaluation of In[677]:= {0.,0.,0.,0.,0.,0.,0.,2.30318381284*10^-9,0.,6.90955143853*10^-9,0.,9.61995269296*10^-9,5.69697254917*10^-7,9.87743724844*10^-9,1.3672734118*10^-6,1.76981031756*10^-8,1.49343004982*10^-6,0.0000564089441028,1.30461472525*10^-6,0.000101522355044,2.77803040269*10^-6,0.000080444698405,0.00279517589237,0.0000608197879561,0.00335427612817,0.000170500598533,0.00165345256966,0.0693649220775,0.00122266773579,0.0417305935458,0.00486469222763,0.00814554217797,0.692325575669,0.0104891127162,0.0048897146358,0.0538440661273,0.00148077578157,0.0895042858466,0.00275230472928,0.000227967410561,0.00558461451971,0.000109034821235,0.00465390209609,0.000172814143821,5.07164370766*10^-6,0.000218100788761,3.24086876983*10^-6,0.000121186008162,4.13978775527*10^-6,4.46896865368*10^-8,3.79008158365*10^-6,3.28806953291*10^-8,1.57920065985*10^-6,3.44083956859*10^-8,0.,2.47139032277*10^-8,0.,8.23796774257*10^-9,0.,0.,0.,0.,0.,0.,0.}

During evaluation of In[677]:= {0.,2.6276417913*10^-10,1.28023911727*10^-8,3.1441216378*10^-10,3.8407173518*10^-8,5.15850133524*10^-10,5.34731082909*10^-8,1.58455004758*10^-6,5.49043522853*10^-8,3.80206810302*10^-6,9.8376012633*10^-8,4.15304072344*10^-6,0.000104675517164,3.62854672792*10^-6,0.000188339532114,7.72501528272*10^-6,0.000149280603238,0.00389362244643,0.000112930238318,0.0046725276082,0.000316309549488,0.0023063885088,0.0774047329677,0.00170811881128,0.0466499269642,0.00677668471315,0.00919583452073,0.645996802126,0.0118261138072,0.00681193987006,0.0601899893402,0.00206878930601,0.0998792329999,0.00383836226103,0.000422941409518,0.00777936345274,0.000202436666432,0.00648296101122,0.00032067065861,0.0000141032691988,0.000404611625896,9.01350947964*10^-6,0.000224889311753,0.0000115120471474,2.4841041572*10^-7,0.0000105393921256,1.8276940003*10^-7,4.39256640586*10^-6,1.91261217945*10^-7,1.79845104264*10^-9,1.3737377571*10^-7,1.38006989711*10^-9,4.579125857*10^-8,1.21270717041*10^-9,0.}

During evaluation of In[677]:= {3.34528274247*10^-9,1.73468577083*10^-7,3.43045647599*10^-6,1.78126343116*10^-7,8.22757025872*10^-6,3.19130770453*10^-7,8.98775868828*10^-6,0.000170089310524,7.85517643323*10^-6,0.000305912923289,0.0000167165526912,0.000242576350162,0.00506618154608,0.000183671994521,0.00607980780256,0.000513781540176,0.00300642321839,0.0840554447009,0.00223110506019,0.0507695757096,0.00881793681269,0.0101295224527,0.603613575418,0.0130061526652,0.00886450023817,0.065503472249,0.00270234721934,0.108462156308,0.00500198737982,0.000687033480414,0.0101222973509,0.000329200186709,0.00843559244134,0.000521033179746,0.0000305200018317,0.000657198653216,0.0000195110786687,0.00036545123372,0.0000249128926341,8.05850296226*10^-7,0.0000228072133743,5.92941711153*10^-7,9.51048289645*10^-6,6.20452868253*10^-7,1.16629362777*10^-8}

During evaluation of In[677]:= {7.89130328169*10^-7,0.000016681600964,0.000252881212855,0.0000145864210549,0.000454573353685,0.000031022471229,0.000360666238107,0.0062829361827,0.000273409964517,0.00754024424973,0.000763480107174,0.00373692672535,0.0894978915393,0.0027802147369,0.0541983663809,0.0109364682763,0.010968068657,0.564794626503,0.0140568028427,0.0109952836409,0.0699249096885,0.00336765739379,0.115486402647,0.00621522393068,0.00102103123626,0.0125536771221,0.000489947480649,0.0104620440197,0.000774587837456,0.0000566422181736,0.000976574614961,0.0000362260046162,0.000543384380967,0.0000462370474296,1.99271236802*10^-6}

During evaluation of In[677]:= {0.0000518414577311,0.000503109638511,0.00751957532076,0.000381953707819,0.0090246886675,0.00106428669011,0.00448452289808,0.0938905651949,0.00334638939414,0.0570306069733,0.0130900472949,0.0117289770542,0.529198751593,0.0150007259605,0.0131619696786,0.0735759755613,0.00405377678648,0.121156480681,0.00745554634111,0.00142348260056,0.0150249590323,0.000684296304377,0.012521894581,0.00108034722126,0.0000946620041171}

During evaluation of In[677]:= {0.00141383871961,0.00523853304996,0.0973723655727,0.00392248282427,0.0593480258782,0.015244653906,0.0124264739085,0.496520342016,0.015856497841,0.0153304809202,0.0765621207482,0.00475207649088,0.12565161287,0.0087049421142,0.00189128026671}

Out[677]= 0.00376821139042*)

I'm sorry. That was not attitute.

You said "unfortunately I cannot provide community with actual code" even after several different people have specifically asked you for that again and again in several different ways.

You then provided one fragment, but said "I have more functions like f[t,d], which means that my overall speed is significantly lower.", It was my mistake when I wrote "show the function f", when I should have written, as others had, "show all the code." It seems it still isn't possible to understand where and why the code is slow or have a chance of providing your "superfast" solution in a single try.

I sincerely hope it works out for you and someone else can provide what you want.

Since you know a great deal about your function f, but this function must remain a secret, take a few minutes, hours or days and construct a different f, one that is different enough that your secret f will not be compromised, but which has enough similarities to the secret f that others are able to test this, see exactly the time behavior you are seeing and perhaps be able to to give you a some concrete advice on what to do.

Until then the problem is in the class "I typed some stuff in, some stuff came out, what do I do?"

Thanks a lot, unfortunately as you point out code runs faster only after the initial run.. unfortunately the implementation that is going to take place (involving java and mathlnk) demands that every time code is executed kernel is quit and all memory is freed. Hence, I have to rum it every time from scratch... imagine that the final product will be dynamic, and variables a1,a2...,t will change maybe 2-3 times per minute. This is why i need to improve speed. Total 4,5 seconds in my pc (i7 3720qm) is not that good... :(

No, no compile..i do not know where should I implement it? Do you mean compile function f?

hello, thanks for replying. I will add a part of my code to illustrate the problem:

ClearAll["Global`*"]

a1 = 0;
a2 = 0;
a3 = 0;
a4 = 0;
b1 = 0;
b2 = 0;
b3 = 0;
b4 = 0;

a = 20.50;
b = 15.;
pa1 = 0.0241637323943662`;
pa2 = 0.014498239436619723`;
pa3 = 0.0027875586854460006`;
pb1 = 0.018726892605633805`;
pb2 = 0.011236135563380286`;
pb3 = 0.00216035798122065`;
p0 = 0.9264270833333333`;

aTotal = 5*a1 + 3*a2 + 1*(a3 + a4);
bTotal = 5*b1 + 3*b2 + 1*(b3 + b4);
difference = aTotal - bTotal;
t = 0;
taf = 100 - t;

pa1 = 0.0241637323943662`;
pa2 = 0.014498239436619723`;
pa3 = 0.0027875586854460006`;
pb1 = 0.018726892605633805`;
pb2 = 0.011236135563380286`;
pb3 = 0.00216035798122065`;
p0 = 0.9264270833333333`;

f[t_, d_] := 
  f[t, d] = 
   If[t > 100, 0, 
    If[d == difference && t == taf, 1, 
     pa1*f[t + 1, d - 5] + pa2*f[t + 1, d - 3] + pa3*f[t + 1, d - 1] +
       pb1*f[t + 1, d + 5] + pb2*f[t + 1, d + 3] + 
      pb3*f[t + 1, d + 1] + p0*f[t + 1, d]]];

InTime = AbsoluteTime[];
expected = Sum[j*f[0, j], {j, -100, 100}]
OutTime = AbsoluteTime[];
OutTime - InTime

Now, I have more functions like f[t,d], which means that my overall speed is significantly lower. I understand that working with the very same algorithm but converted to serial (done it in excel) runs instantly. Is there any way I can speed up in mathematica while forking on such recursive functions?

Thank you in advance.