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
Keywords
Examples
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.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
- Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018
Crossrefs
Programs
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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 *)
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
Comments