# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a257628 Showing 1-1 of 1 %I A257628 #36 Mar 07 2021 17:38:23 %S A257628 0,1,1,0,0,-1,0,-1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,-1,0,0,0,-1,0,0,0,0,0, %T A257628 0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0, %U A257628 0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0 %N A257628 Expansion of 1 - f(-x) in powers of x where f() is a Ramanujan theta function. %C A257628 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). %H A257628 G. C. Greubel, Table of n, a(n) for n = 0..10000 %H A257628 Michael Somos, Introduction to Ramanujan theta functions %H A257628 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions %H A257628 Wikipedia, Pentagonal number theorem %F A257628 G.f.: x + x^2 * (1 - x) + x^3 * (1 - x) * (1 - x^2) + .... %F A257628 G.f.: Sum_{k>0} -(-1)^k * (x^((3*k^2 - k)/2) + x^((3*k^2 + k)/2)). %F A257628 G.f.: Sum_{k>0} -(-1)^k * x^((k^2 + k) / 2) / ((1 - x) * (1 - x^2) * ... * (1 - x^k)). %F A257628 G.f.: -(Product_{j>=1}(1-x^j) - 1), from Euler's Pentagonal Theorem. - _Wolfdieter Lang_, Feb 16 2021 %F A257628 a(n) = - A010815(n) unless n=0, a(0) = 0. %e A257628 G.f. = x + x^2 - x^5 - x^7 + x^12 + x^15 - x^22 - x^26 + x^35 + x^40 + ... %e A257628 G.f. = q^25 + q^49 - q^121 - q^169 + q^289 + q^361 - q^529 - q^625 + ... %t A257628 a[ n_] := SeriesCoefficient[ 1 - QPochhammer[ x], {x, 0, n}]; %t A257628 a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ n > 0 && IntegerQ[m], - KroneckerSymbol[ 12, m], 0]]; %o A257628 (PARI) {a(n) = if( n<0, 0, polcoeff( 1 - eta(x + x * O(x^n)), n))}; %o A257628 (PARI) {a(n) = my(m); if( n>0 && issquare( 24*n + 1, &m), - kronecker( 12, m))}; %Y A257628 Cf. A001318, A010815, A341418 (convolution triangle). %K A257628 sign,easy %O A257628 0 %A A257628 _Michael Somos_, Jul 12 2015 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE