A287990 - OEIS

%I #35 May 17 2022 04:45:28

%S 1,36,630,7176,60165,398412,2184078,10255320,42321942,156590980,

%T 527649912,1639560888,4745867595,12904341336,33190117110,81222775680,

%U 190066236318,427113304920,925107172122,1937505253320,3934709716500,7767340567380,14937197788890

%N Expansion of Jacobi theta constant (theta_2/2)^36.

%C Number of ways of writing n as the sum of 36 triangular numbers.

%H Seiichi Manyama, <a href="/A287990/b287990.txt">Table of n, a(n) for n = 0..10000</a>

%H Simon Plouffe, <a href="http://vixra.org/abs/1409.0048">Conjectures of the OEIS, as of June 20, 2018</a>

%F a(0) = 1, a(n) = (36/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0.

%F a(n) = 1/141557760 * Sum_{a, b, c, x, y, z > 0, a*x + b*y + c*z = 2*n + 9, a == b == c == x == y == z == 1 mod 2 and a > b > c} a*b*c*((a^2 - b^2)*(a^2 - c^2)*(b^2 - c^2))^2.

%F Euler transform of [36, -36, 36, -36, 36, -36, ...]. - _Simon Plouffe_, Jun 23 2018

%e 5*1 + 3*1 + 1*3 = 7*1 + 3*1 + 1*1 = 2 + 9. So a(1) = (5*3*1*((25-9)*(25-1)*(9-1))^2 + 7*3*1*((49-9)*(49-1)*(9-1))^2) / 141557760 = 36.

%p a:= proc(n) option remember; `if`(n=0, 1, -add(a(n-j)*add(

%p       36*d*(-1)^d, d=numtheory[divisors](j)), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 23 2018

%t A002129[n_] := DivisorSum[n, -(-1)^#*#&];

%t a[n_] := a[n] = If[n == 0, 1, (36/n)*Sum[A002129[k]*a[n-k], {k, 1, n}]];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 17 2022 *)

%Y Column k=36 of A286180.

%Y Cf. A008438 (k=4*1^2), A014805 (k=4*2^2), this sequence (k=4*3^2).

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jun 04 2017