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A255065 Expansion of x * psi(x^5) * f(-x^10) / f(-x^4, -x^6) in powers of x where psi(), f() are Ramanujan theta functions.

%I A255065 #28 Feb 16 2025 08:33:25
%S A255065 1,0,0,0,1,1,1,0,1,1,1,1,2,1,2,2,3,2,3,3,4,4,4,4,6,5,6,6,8,8,10,9,11,
%T A255065 11,13,13,16,15,17,18,21,21,24,24,28,29,32,33,38,38,43,44,49,51,57,58,
%U A255065 65,67,73,76,85,87,95,99,109,113,123,127,139,145,157
%N A255065 Expansion of x * psi(x^5) * f(-x^10) / f(-x^4, -x^6) in powers of x where psi(), f() are Ramanujan theta functions.
%C A255065 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A255065 Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).
%D A255065 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 7th equation.
%H A255065 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A255065 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A255065 Expansion of  x * f(-x, -x^9) * f(-x^10) / f(-x, -x^4) in powers of x where f(,) is the Ramanujan general theta function.
%F A255065 Expansion of x * psi(x^5) * H(x^2) in powers of x where f(,) is the Ramanujan general theta function and H() is a Rogers-Ramanujan function. - _Michael Somos_, Jul 09 2015
%F A255065 Euler transform of period 10 sequence [ 0, 0, 0, 1, 1, 1, 0, 0, 0, -1, ...].
%F A255065 G.f.: x * (Sum_{k>0} x^(5*k*(k-1)/2)) / (Product_{k in Z} 1 - x^abs(10*k + 4)).
%F A255065 A053266(n) = A053264(n) + a(n) unless n=0.
%e A255065 G.f. = x + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + x^12 + 2*x^13 + x^14 + ...
%e A255065 G.f. = q^119 + q^599 + q^719 + q^839 + q^1079 + q^1199 + q^1319 + q^1439 + ...
%t A255065 a[ n_] := SeriesCoefficient[ Product[ x * (1 - x^k)^{ 0, 0, 0, -1, -1, -1, 0, 0, 0, 1} [[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];
%o A255065 (PARI) {a(n) = if( n<1, 0, n--; polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, 0, 0, 0, -1, -1, -1, 0, 0, 0][k%10+1]), n))};
%Y A255065 Cf. A053264, A053266.
%K A255065 nonn
%O A255065 1,13
%A A255065 _Michael Somos_, Jul 08 2015