# How to accelerate combinations and sum calculations here?

I have a 4d matrix H defined as below (embedded with combinations and sums) H[n1_, n2_, n3_, n4_] :=
(1/(4*Sqrt[n1*(n1 + 1)*n2*(n2 + 1)*n3*(n3 + 1)*n4*(n4 + 1)]))*
Sum[(-1)^(m1 + m2 + m3 + m4)*Binomial[n1 + 1, m1 + 2]*Binomial[n2 + 1, m2 + 2]*Binomial[n3 + 1, m3 + 2]*Binomial[n4 + 1, m4 + 2]*
Binomial[m1 + m3, m1]*Binomial[m2 + m4, m2]*(m1 + m3 + 1)*(m2 + m4 + 1)*
(((m1 + m3 + 2)/2^(m1 + m3))*Sum[Binomial[m1 + m3 + k + 2, k]/2^k, {k, 0, m2 + m4 + 1}] +
((m2 + m4 + 2)/2^(m2 + m4))*Sum[Binomial[m2 + m4 + l + 2, l]/2^l, {l, 0, m1 + m3 + 1}]), {m1, 0, n1 - 1}, {m2, 0, n2 - 1}, {m3, 0, n3 - 1},
{m4, 0, n4 - 1}]


Here n1, n2, n3, n4 are indices that range from 1 to N.

When N = 6, it takes my Mac (2.7 GHz Intel Core i5) about 30s to figure out the whole 4d matrix. However, it takes 1.5hrs for N=11. The time scaling is highly nonlinear..

The problem is that when N is big (N>20), it takes forever to run. Anyone has some good suggestion to accelerate the calculation? Thanks.

UPDATE: Actually I do not need all the matrix entries, please see here:

Delete rows and columns in a matrix based on the element index

After getting and 4d matrix H and reshaping it into a 2d matrix, I will delete according to the rule:

If i>j OR k>l, then delete this element.

So it seems redundant to figure all the entries out in the first place?

• I guess I should have tried ParallelTable and let's see what happens. – James LT May 12 '16 at 17:00
• James, do you need symbolic results? Numerical results are likely to be calculated faster. Also, you could try ParallelSum instead of Sum in your definition of H. See also my alternative proposal in my answer below. – MarcoB May 12 '16 at 18:33
• @MarcoB Thanks for the advice. Now I cut the time by more than 50% combining all the suggestions in the post. – James LT May 14 '16 at 19:25

You could introduce further conditional definitions for H which will prevent those computations whose results would end up being thrown away.

H[i_, j_, k_, l_] /; (i > j || k > l) = Missing[];

m = Table[H[n, 2, 3, 4], {n, 1, 10}]