cp's OEIS Frontend

A207641 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1+x^k)/(1-x^k).

%I A207641 #36 Feb 16 2025 08:33:16
%S A207641 1,1,3,5,9,15,25,39,61,93,139,205,299,429,611,861,1201,1663,2285,3115,
%T A207641 4221,5683,7605,10123,13405,17661,23163,30245,39323,50925,65699,84445,
%U A207641 108167,138089,175719,222921,281965,355627,447309,561139,702133,876395,1091301
%N A207641 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1+x^k)/(1-x^k).
%C A207641 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A207641 In Ramanujan's equation let a = x and b = 1. - _Michael Somos_, Nov 20 2015
%D A207641 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 370, 9th equation.
%H A207641 Seiichi Manyama, <a href="/A207641/b207641.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..750 from Vaclav Kotesovec)
%H A207641 Frank Garvan, <a href="http://www.combinatorics.net/conf/A75/Slides/02_03_Garvan.pdf">Dyson's rank function and Andrews's SPT-function</a>, slides 29, 30.
%H A207641 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A207641 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A207641 Expansion of 1 / ((1 + x) * phi(-x)) in powers of x where phi() is a Ramanujan theta function. - _Michael Somos_, Nov 20 2015
%F A207641 G.f.: 1 + x*(1+x) * (1 / (1-x)^2 + 2*x^3 / ((1-x)*(1-x^2))^2 + 2*x^7*(1+x) / ((1-x)*(1-x^2)*(1-x^3))^2 + 2*x^12*(1+x)*(1+x^2) / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4))^2 + ...). [Ramanujan] - _Michael Somos_, Nov 20 2015
%F A207641 a(n) + a(n+1) = A015128(n+1) for n >= 0. - _Seiichi Manyama_, Jul 12 2018
%F A207641 a(n) ~ exp(Pi*sqrt(n)) / (16*n). - _Vaclav Kotesovec_, Jun 18 2019
%e A207641 G.f.: A(x) = 1 + x + 3*x^2 + 5*x^3 + 9*x^4 + 15*x^5 + 25*x^6 + 39*x^7 +...
%e A207641 such that, by definition,
%e A207641 A(x) = 1 + x*(1+x)/(1-x) + x^2*(1+x)*(1+x^2)/((1-x)*(1-x^2)) + x^3*(1+x)*(1+x^2)*(1+x^3)/((1-x)*(1-x^2)*(1-x^3)) +...
%t A207641 a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {-x}, {}, x, x], {x, 0, n}]; (* _Michael Somos_, Mar 11 2014 *)
%t A207641 a[ n_] := SeriesCoefficient[ 1 / ((1 + x) EllipticTheta[ 4, 0, x]), {x, 0, n}]; (* _Michael Somos_, Nov 20 2015 *)
%o A207641 (PARI) {a(n)=polcoeff(sum(m=0,n,x^m*prod(k=1,m,(1+x^k)/(1-x^k +x*O(x^n))) ),n)}
%o A207641 for(n=0,50,print1(a(n),", "))
%o A207641 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / ((1 + x) * eta(x + A)^2), n))}; /* _Michael Somos_, Nov 20 2015 */
%Y A207641 Cf. A015128, A059777.
%K A207641 nonn
%O A207641 0,3
%A A207641 _Paul D. Hanna_, Feb 19 2012