A092876 - OEIS

%I #25 Mar 12 2021 22:24:42

%S 1,-1,1,-2,1,0,1,1,-1,-2,0,0,2,2,-3,2,-6,3,1,2,2,-2,-4,0,-2,5,7,-8,6,

%T -16,7,1,6,6,-7,-10,1,-2,11,14,-17,12,-34,16,3,12,11,-12,-22,1,-6,24,

%U 30,-36,25,-70,32,6,25,24,-26,-42,2,-10,45,56,-68,48,-132,60,12,45,43,-46,-78,4,-22,84,106,-126,89,-242,110,20,84,80

%N Expansion of q * (f(-q, -q^12) * f(-q^3, -q^10) * f(-q^4, -q^9)) / (f(-q^2, -q^11) * f(-q^5, -q^8) * f(-q^6, -q^7)) in powers of q where f(, ) is Ramanujan's general theta function.

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

%H G. C. Greubel, <a href="/A092876/b092876.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%F Euler transform of period 13 sequence [-1, 1, -1 ,-1, 1, 1 ,1, 1, -1, -1, 1, -1, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v + u*v^3 + u^3*v^2 + 2*u*v * (1 - u + v + u*v).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3*v * (3 + 3*v + v^2) + 3*u^2*v * (v^2 + v - 1) + u*v * (1 - 3*v + 3*v^2) - (u^4 + v^4)

%F G.f.: x * Product_{k>0} (1 - x^k)^Kronecker(13, k).

%F a(n) = A214157(n) - A133099(n) unless n=0. - _Michael Somos_, Jul 05 2012

%F Convolution inverse is A214157.

%e G.f. = q - q^2 + q^3 - 2*q^4 + q^5 + q^7 + q^8 - q^9 - 2*q^10 + 2*q^13 + 2*q^14 + ...

%t a[ n_] := SeriesCoefficient[ q Product[ (1 - q^k)^KroneckerSymbol[ 13, k], {k, n - 1}], {q, 0, n}]; (* _Michael Somos_, Jan 17 2015 *)

%o (PARI) {a(n) = if( n<1, 0, n--; polcoeff( prod( k=1, n, (1 - x^k)^kronecker( 13, k), 1 + x * O(x^n)), n))}; /* _Michael Somos_, Oct 24 2005 */

%o (PARI) {a(n) = my(A, u, v); if( n<0, 0, A = x; for( k=2, n, u = A + x * O(x^k); v = subst(u, x, x^2); A -= x^k * polcoeff( u^2 - v + u*v^3 + u^3*v^2 + 2*u*v * (1 - u + v + u*v), k+1) / 2); polcoeff(A, n))};

%Y Cf. A133099, A214157.

%K sign

%O 1,4

%A _Michael Somos_, Mar 09 2004