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A340949 Number of ways to write n as an ordered sum of 4 nonzero triangular numbers.

%I A340949 #8 Jan 31 2021 20:17:14
%S A340949 1,0,4,0,6,4,4,12,1,16,6,16,12,12,22,8,36,4,30,24,21,36,18,36,28,48,
%T A340949 16,44,36,44,48,36,46,40,72,20,73,48,54,72,42,68,56,84,50,72,78,56,84,
%U A340949 84,62,112,60,60,110,84,97,72,120,76,116,84,72,144,102,104,96,108,102,156,102,92
%N A340949 Number of ways to write n as an ordered sum of 4 nonzero triangular numbers.
%H A340949 Alois P. Heinz, <a href="/A340949/b340949.txt">Table of n, a(n) for n = 4..10000</a>
%F A340949 G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^4, where theta_2() is the Jacobi theta function.
%p A340949 b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
%p A340949       if n=0 then `if`(k=0, 1, 0) else
%p A340949       while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
%p A340949     end:
%p A340949 a:= n-> b(n, 4):
%p A340949 seq(a(n), n=4..75);  # _Alois P. Heinz_, Jan 31 2021
%t A340949 nmax = 75; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^4, {x, 0, nmax}], x] // Drop[#, 4] &
%Y A340949 Cf. A000217, A008438, A010054, A053603, A053604, A319814, A340950, A340951, A340952, A340953, A340954, A340955.
%K A340949 nonn
%O A340949 4,3
%A A340949 _Ilya Gutkovskiy_, Jan 31 2021