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A340951 Number of ways to write n as an ordered sum of 6 nonzero triangular numbers.

%I A340951 #8 Jan 31 2021 20:19:54
%S A340951 1,0,6,0,15,6,20,30,15,66,21,90,61,90,126,86,210,90,270,156,261,320,
%T A340951 210,450,261,516,375,542,495,570,727,540,870,650,966,816,1050,906,
%U A340951 1155,1266,1020,1560,1090,1710,1416,1698,1635,1746,2120,1650,2376,1980,2316,2490,2368,2520,2835
%N A340951 Number of ways to write n as an ordered sum of 6 nonzero triangular numbers.
%H A340951 Alois P. Heinz, <a href="/A340951/b340951.txt">Table of n, a(n) for n = 6..10000</a>
%F A340951 G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^6, where theta_2() is the Jacobi theta function.
%p A340951 b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
%p A340951       if n=0 then `if`(k=0, 1, 0) else
%p A340951       while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
%p A340951     end:
%p A340951 a:= n-> b(n, 6):
%p A340951 seq(a(n), n=6..62);  # _Alois P. Heinz_, Jan 31 2021
%t A340951 nmax = 62; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^6, {x, 0, nmax}], x] // Drop[#, 6] &
%Y A340951 Cf. A000217, A008440, A010054, A053603, A053604, A319816, A340949, A340950, A340952, A340953, A340954, A340955.
%K A340951 nonn
%O A340951 6,3
%A A340951 _Ilya Gutkovskiy_, Jan 31 2021