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A340954 Number of ways to write n as an ordered sum of 9 nonzero triangular numbers.

%I A340954 #8 Jan 31 2021 20:15:30
%S A340954 1,0,9,0,36,9,84,72,126,261,162,576,336,882,873,1092,1845,1386,3061,
%T A340954 2160,4167,3957,4860,6948,5580,10287,7812,12777,12276,14634,18363,
%U A340954 17136,25056,21282,31266,28899,36075,39654,41202,51348,49383,63270,59391,76059,73611,87319,93582,96966
%N A340954 Number of ways to write n as an ordered sum of 9 nonzero triangular numbers.
%H A340954 Alois P. Heinz, <a href="/A340954/b340954.txt">Table of n, a(n) for n = 9..10000</a>
%F A340954 G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^9, where theta_2() is the Jacobi theta function.
%p A340954 b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
%p A340954       if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
%p A340954       while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
%p A340954     end:
%p A340954 a:= n-> b(n, 9):
%p A340954 seq(a(n), n=9..56);  # _Alois P. Heinz_, Jan 31 2021
%t A340954 nmax = 56; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^9, {x, 0, nmax}], x] // Drop[#, 9] &
%Y A340954 Cf. A000217, A010054, A053603, A053604, A226253, A319819, A340949, A340950, A340951, A340952, A340953, A340955.
%K A340954 nonn
%O A340954 9,3
%A A340954 _Ilya Gutkovskiy_, Jan 31 2021