[?]  Can this expression be written in closed form?

Questions
1

Observe the following Wolfram Mathematica expression which results in a table of integer sequences:

b[n_, k_] := b[n, k] = If[n == 0, 1, If[n == 1, If[k > 0, 1, 0], Sum[Binomial[n-1, r-1]*b[r-1, k-1]*b[n-r, k-1], {r, 1, n}] ] ]; t [n_, k_] := b[n, k] - If[k > 0, b[n, k-1], 0]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}]

I’m struggling to interpret what this is actually doing. Clearly it’s quite a complicated mess of if statements, but I’m not entirely sure if they are necessary or not. I obtained the expression from here:

Because I wanted to determine an expression (dependent on the two variables n and k) which would result in table of integers seen there. Would somebody please write the Mathematica code above in closed form so that it’s easier to understand? If this is not possible, can you at least try to explain what’s going on, and perhaps give an expression which models the above in most situations?

2

The code isn’t pretty. That’s mostly because it’s been directly translated from the Maple code.

The first step is to write this out as a recurrence equation. Once you’ve done that you can use RSolve to see if Mathematica knows a closed form for the reccurence you’re looking at. For more examples, see the documentation under “tutorial/SolvingRecurrenceEquations”.

3

Hi John,

I would write the code generating this sequence like so:

ClearAll["Global`*"]

b[0, _] = 1;
b[1, 0] = 0;
b[1, k_] := If[k > 0, 1, 0];
b[n_, k_] := b[n, k] = Sum[Binomial[n - 1, r - 1]*b[r - 1, k - 1]*b[n - r, k - 1], {r, 1, n}]
t[n_, k_] := b[n, k] - b[n, k - 1]

sequence = Flatten@Table[t[n, k], {n, 0, 10}, {k, 0, n}]

which - I think - is much easier to read (even though there surely might be simpler ways!). As a brute force method for getting a closed form FindSequenceFunction can be helpful - but unfortunately not here! Maybe one gets a little “understanding” on what is going on with this:

logData = Flatten[Table[{n, k, Log[1. + t[n, k]]}, {n, 0, 10}, {k, 0, n}], 1];
route = Flatten[Table[{n, k, 14}, {n, 0, 10}, {k, 0, n}], 1];
Show[ListPlot3D[logData, ImageSize -> Large], Graphics3D[{Thick, Line[route]}]]

which gives:

sequence.png
sequence.png576×455 76.9 KB

Hope this helps a little, regards – Henrik