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A259771 Expansion of x * psi(x^5) * f(-x^10) / f(-x^2,-x^8) in powers of x where psi(), f() are Ramanujan theta functions.

%I A259771 #15 Feb 16 2025 08:33:26
%S A259771 1,0,1,0,1,1,1,1,2,1,2,1,3,2,3,3,4,4,5,4,6,5,7,7,9,8,10,10,12,12,15,
%T A259771 14,18,17,20,20,24,24,28,28,33,33,38,38,44,45,50,52,59,60,68,69,78,80,
%U A259771 89,92,102,105,116,120,133,137,151,156,171,178,194,201
%N A259771 Expansion of x * psi(x^5) * f(-x^10) / f(-x^2,-x^8) in powers of x where psi(), f() are Ramanujan theta functions.
%C A259771 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A259771 The g.f. for this sequence is the last term of the 14th equation on page 20 of Ramanujan 1988.
%D A259771 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 20
%H A259771 G. C. Greubel, <a href="/A259771/b259771.txt">Table of n, a(n) for n = 1..1000</a>
%H A259771 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A259771 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A259771 Euler transform of period 10 sequence [ 0, 1, 0, 0, 1, 0, 0, 1, 0, -1, ...].
%F A259771 a(n) = A053265(n-1) - A053267(n).
%e A259771 G.f. = x + x^3 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + x^10 + 2*x^11 + x^12 + ...
%e A259771 G.f. = q^49 + q^289 + q^529 + q^649 + q^769 + q^889 + 2*q^1009 + q^1129 + ...
%t A259771 a[ n_] := SeriesCoefficient[ x Product[ (1 - x^k)^{ 0, -1, 0, 0, -1, 0, 0, -1, 0, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];
%t A259771 QP:= QPochhammer; a[n_]:= SeriesCoefficient[ x*QP[x^10]/(QP[x^5, x^10]* QP[x^2, x^10]*QP[x^8, x^10]), {x, 0, n}]; Table[a[n], {n, 1, 100}] (* _G. C. Greubel_, Mar 16 2018 *)
%o A259771 (PARI) {a(n) = if( n<1, 0, n--; polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, 0, -1, 0, 0, -1, 0, 0, -1, 0][k%10 + 1]), n))};
%Y A259771 Cf. A053265, A053267.
%K A259771 nonn
%O A259771 1,9
%A A259771 _Michael Somos_, Jul 04 2015