Hi Parminder,

Using WL

b[p_Integer, n_Integer] := (-Power[-1, 
      n/Power[2, p - 1] + Sum[-b[p - i, n]/Power[2, i], {i, 1, p - 1}]] + 1)/2

$$\frac{1}{2} \left(-(-1)^{\frac{n}{2^{p-1}}+\sum _{i=1}^{p-1} -\frac{b(p-i,n)}{2^i}}+1\right)$$

b[2, 7]
(* 1 *)

Table[b[p, n], {p, 0, 7}, {n, 0, 80}] // ArrayPlot

I don't know how to do this on Wolfram|Alpha.

Parminder,

You can sign up for a free Wolfram Cloud account here and use the WL from a browser (there are some resource consumption limitations).

If you are unfamiliar with WL then take a look at Stephen Wolfram's book which is available online for free.

You might also find this helpful.

Have fun!