A178737 - OEIS

%I #15 Sep 19 2018 00:56:22

%S 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,

%T 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,

%U 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

%N Coefficients in expansion of Jacobi theta_1'''(0).

%H G. C. Greubel, <a href="/A178737/b178737.txt">Table of n, a(n) for n = 0..5000</a>

%F 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).

%F 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).

%F Convolution of A006352 and A010816.

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

%e G.f. = 1 - 27*x + 125*x^3 - 343*x^6 + 729*x^10 - 1331*x^15 + 2197*x^21 + ...

%e G.f. = q - 27*q^9 + 125*q^25 - 343*q^49 + 729*q^81 - 1331*q^121 + ...

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

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

%Y Cf. A006352, A010816.

%K sign

%O 0,2

%A _Michael Somos_, Jun 08 2010