A128617 - OEIS

%I #17 Feb 16 2025 08:33:05

%S 0,1,1,0,1,0,0,1,0,0,0,1,0,0,0,0,2,1,0,1,0,0,2,0,0,0,1,0,0,1,0,1,0,0,

%T 0,0,0,2,0,0,0,0,0,0,1,0,2,1,0,1,0,0,2,0,0,0,2,0,0,0,0,2,0,0,0,0,0,2,

%U 0,0,0,1,0,0,1,0,0,0,0,1,0,0,2,0,0,0,0

%N Expansion of q^2 * psi(q) * psi(q^15) in powers of q where psi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Also the number of positive odd solutions to equation x^2 + 15y^2 = 8n. - _Seiichi Manyama_, May 21 2017

%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(i).

%H Seiichi Manyama, <a href="/A128617/b128617.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of (eta(q^2) * eta(q^30))^2 / (eta(q) * eta(q^15)) in powers of q.

%F Euler transform of period 30 sequence [ 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 2, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, ...].

%F For n>0, n in A028955 equivalent to a(n) nonzero. If a(n) nonzero, a(n) = A082451(n) and a(n) = -A121362(n).

%F a(n)= (A082451(n) - A121362(n) )/2.

%F G.f.: x^2 * Product_{k>0} (1 - x^k) * (1 - x^(15*k)) * (1 + x^(2*k))^2 * (1 + x^(30*k))^2.

%e G.f. = x^2 + x^3 + x^5 + x^8 + x^12 + 2*x^17 + x^18 + x^20 + 2*x^23 + x^27 + x^30 + ...

%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -60, #] - KroneckerSymbol[ 20, #] KroneckerSymbol[ -3, n/#] &] / 2]; (* _Michael Somos_, Nov 12 2015 *)

%t a[ n_] := SeriesCoefficient[ q^2 (QPochhammer[ q^2] QPochhammer[ q^30])^2 / (QPochhammer[ q] QPochhammer[ q^15]), {q, 0, n}]; (* _Michael Somos_, Nov 12 2015 *)

%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-60, d) - kronecker(20, d) * kronecker(-3, n/d) )/2)};

%o (PARI) {a(n) = my(A); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^30 + A))^2 / (eta(x + A) * eta(x^15 + A)), n))};

%Y Cf. A035162.

%K nonn

%O 1,17

%A _Michael Somos_, Mar 13 2007