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A340955 Number of ways to write n as an ordered sum of 10 nonzero triangular numbers.

%I A340955 #7 Jan 31 2021 20:25:33
%S A340955 1,0,10,0,45,10,120,90,210,370,297,930,570,1620,1480,2220,3375,2940,
%T A340955 6085,4590,8981,8370,11430,15100,13890,23832,19155,31940,30195,38520,
%U A340955 46890,46440,66550,59400,86355,81532,104220,114390,122410,153450,149490,193440,188010,235350,238840
%N A340955 Number of ways to write n as an ordered sum of 10 nonzero triangular numbers.
%H A340955 Alois P. Heinz, <a href="/A340955/b340955.txt">Table of n, a(n) for n = 10..10000</a>
%F A340955 G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^10, where theta_2() is the Jacobi theta function.
%p A340955 b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
%p A340955       if n=0 then `if`(k=0, 1, 0) else
%p A340955       while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
%p A340955     end:
%p A340955 a:= n-> b(n, 10):
%p A340955 seq(a(n), n=10..54);  # _Alois P. Heinz_, Jan 31 2021
%t A340955 nmax = 54; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^10, {x, 0, nmax}], x] // Drop[#, 10] &
%Y A340955 Cf. A000217, A010054, A053603, A053604, A226254, A319820, A340949, A340950, A340951, A340952, A340953, A340954.
%K A340955 nonn
%O A340955 10,3
%A A340955 _Ilya Gutkovskiy_, Jan 31 2021