A287990 Expansion of Jacobi theta constant (theta_2/2)^36.

1, 36, 630, 7176, 60165, 398412, 2184078, 10255320, 42321942, 156590980, 527649912, 1639560888, 4745867595, 12904341336, 33190117110, 81222775680, 190066236318, 427113304920, 925107172122, 1937505253320, 3934709716500, 7767340567380, 14937197788890

Offset

0,2

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018

Formula

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

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.

Euler transform of [36, -36, 36, -36, 36, -36, ...]. - Simon Plouffe, Jun 23 2018

Example

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.

Maple

a:= proc(n) option remember; `if`(n=0, 1, -add(a(n-j)*add(
      36*d*(-1)^d, d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 23 2018

Mathematica

A002129[n_] := DivisorSum[n, -(-1)^#*#&];
a[n_] := a[n] = If[n == 0, 1, (36/n)*Sum[A002129[k]*a[n-k], {k, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 17 2022 *)

Crossrefs

Column k=36 of A286180.

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

Sequence in context: A210432 A010952 A323980 * A283630 A035713 A035478

Adjacent sequences: A287987 A287988 A287989 * A287991 A287992 A287993

Keyword

nonn

Author

Seiichi Manyama, Jun 04 2017

Status

approved