A178737 Coefficients in expansion of Jacobi theta_1'''(0).

1, -27, 0, 125, 0, 0, -343, 0, 0, 0, 729, 0, 0, 0, 0, -1331, 0, 0, 0, 0, 0, 2197, 0, 0, 0, 0, 0, 0, -3375, 0, 0, 0, 0, 0, 0, 0, 4913, 0, 0, 0, 0, 0, 0, 0, 0, -6859, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9261, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12167, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15625, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Michael Somos, Jun 08 2010

Keywords

• sign

Examples

			G.f. = 1 - 27*x + 125*x^3 - 343*x^6 + 729*x^10 - 1331*x^15 + 2197*x^21 + ...
G.f. = q - 27*q^9 + 125*q^25 - 343*q^49 + 729*q^81 - 1331*q^121 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A006352, A010816.

Programs

Mathematica

a[ n_] := With[ {x = Sqrt[8 n + 1]}, If[ IntegerQ[x], (-1)^Quotient[x, 2] x^3, 0]]; (* Michael Somos, Mar 19 2017 *)

PARI

{a(n) = my(x); if( n<0, 0, if(issquare(8*n + 1, &x), (-1)^(x\2) * x^3))};

Formula

Given g.f. A(x), then q * A(q^8) = -1/2 * theta_1'''(0, q^4) where the Jacobi nome q = exp(-Pi * K' / K).

a(n) = b(8*n + 1) where b() is multiplicative with b(p^e) = 0 if e odd, b(2^e) = 0^e, b(p^e) = p^(3 * e/2) if p == 1 (mod 4), b(p^e) = (-p)^(3 * e/2) if p == 3 (mod 4).

Convolution of A006352 and A010816.

G.f.: Sum_{k>=0} (-1)^k * (2*k + 1)^3 * x^(k * (k+1) / 2).