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A340953 Number of ways to write n as an ordered sum of 8 nonzero triangular numbers.

%I A340953 #8 Jan 31 2021 20:24:10
%S A340953 1,0,8,0,28,8,56,56,70,176,84,336,196,448,492,504,953,616,1456,960,
%T A340953 1814,1792,1904,3032,2100,4144,3052,4768,4670,5264,6720,5936,8876,
%U A340953 7112,10620,9648,11718,12720,13216,15960,15261,19608,17164,23296,21226,25424,26796,27272,32844
%N A340953 Number of ways to write n as an ordered sum of 8 nonzero triangular numbers.
%H A340953 Alois P. Heinz, <a href="/A340953/b340953.txt">Table of n, a(n) for n = 8..10000</a>
%F A340953 G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^8, where theta_2() is the Jacobi theta function.
%p A340953 b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
%p A340953       if n=0 then `if`(k=0, 1, 0) else
%p A340953       while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
%p A340953     end:
%p A340953 a:= n-> b(n, 8):
%p A340953 seq(a(n), n=8..56);  # _Alois P. Heinz_, Jan 31 2021
%t A340953 nmax = 56; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^8, {x, 0, nmax}], x] // Drop[#, 8] &
%Y A340953 Cf. A000217, A007331, A010054, A053603, A053604, A319818, A340949, A340950, A340951, A340952, A340954, A340955.
%K A340953 nonn
%O A340953 8,3
%A A340953 _Ilya Gutkovskiy_, Jan 31 2021