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A287991 Expansion of Jacobi theta constant (theta_2/2)^48.

%I A287991 #28 Apr 22 2021 11:22:03
%S A287991 1,48,1128,17344,196836,1764192,13051008,82244736,452197434,
%T A287991 2210431056,9753024192,39328459968,146436844568,507826976160,
%U A287991 1652238451200,5074887938688,14794635174459,41126600601168,109456398969568,279899944411776,689873759134308
%N A287991 Expansion of Jacobi theta constant (theta_2/2)^48.
%C A287991 Number of ways of writing n as the sum of 48 triangular numbers.
%H A287991 Seiichi Manyama, <a href="/A287991/b287991.txt">Table of n, a(n) for n = 0..10000</a>
%F A287991 a(0) = 1, a(n) = (48/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0.
%F A287991 a(n) = 1/3110400 * Sum_{a, b, c, x, y, z > 0, a*x + b*y + c*z = n + 6, x == y == z == 1 mod 2 and a > b > c} (a*b*c)^3*((a^2 - b^2)*(a^2 - c^2)*(b^2 - c^2))^2.
%F A287991 G.f.: exp(48*Sum_{k>=1} (x^k/k)/(1 + x^k)). - _Ilya Gutkovskiy_, Aug 02 2017
%e A287991 4*1 + 2*1 + 1*1 = 1 + 6. So a(1) = (4*2*1)^3*((16-1)*(16-4)*(4-1))^2 / 3110400 = 48.
%t A287991 a002129[n_]:=-Sum[(-1)^d*d, {d, Divisors[n]}]; a[n_]:=a[n]=If[n==0, 1, 48 Sum[a002129[k] a[n - k], {k, n}]/n]; Table[a[n], {n, 0, 100}] (* _Indranil Ghosh_, Aug 02 2017 *)
%o A287991 (Python)
%o A287991 from sympy import divisors
%o A287991 from sympy.core.cache import cacheit
%o A287991 def a002129(n): return -sum((-1)**d*d for d in divisors(n))
%o A287991 @cacheit
%o A287991 def a(n): return 1 if n==0 else 48*sum(a002129(k)*a(n - k) for k in range(1, n + 1))//n
%o A287991 print([a(n) for n in range(101)]) # _Indranil Ghosh_, Aug 02 2017
%Y A287991 Column k=48 of A286180.
%Y A287991 Cf. A007331 (k=4*1*2), A014809 (k=4*2*3), this sequence (k=4*3*4).
%K A287991 nonn
%O A287991 0,2
%A A287991 _Seiichi Manyama_, Jun 04 2017