A006305 - OEIS

%I M1014 #28 Dec 18 2021 22:17:12

%S 1,2,4,6,10,16,25,38,58,84,122,174,244,338,465,630,850,1136,1508,1988,

%T 2608,3398,4408,5688,7306,9342,11900,15090,19070,24008,30122,37666,

%U 46955,58348,72302,89338,110094,135316,165912,202924,247632,301508

%N Taylor series related to one in Ramanujan's Lost Notebook.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vaclav Kotesovec, <a href="/A006305/b006305.txt">Table of n, a(n) for n = 0..20000</a>

%H G. E. Andrews, <a href="http://dx.doi.org/10.1007/BFb0096452">Mordell integrals and Ramanujan's "Lost" Notebook</a>, pp. 10-48 of Analytic Number Theory (Philadelphia 1980), Lect. Notes Math. 899 (1981).

%F G.f.: Sum_{n>=0} q^(n^2+n) (1+q^2)(1+q^4)...(1+q^(2n))/((1-q)^2 (1-q^2) (1-q^3)^2 (1-q^4) ... (1-q^(2n)) (1-q^(2n+1))^2).

%F a(n) ~ c * exp(r*sqrt(n)) / n^(3/4), where r = 2.74858241446108527... and c = 0.1051685561271293027... - _Vaclav Kotesovec_, Jun 12 2019

%e G.f. = 1 + 2*x + 4*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 25*x^6 + 38*x^7 + 58*x^8 + ...

%t Series[Sum[q^(n^2+n)/(1-q)^2 Product[(1+q^(2k))/((1-q^(2k))(1-q^(2k+1))^2), {k, 1, n}], {n, 0, 9}], {q, 0, 100}]

%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k k + k) QPochhammer[ -x^2, x^2, k] / (QPochhammer[ x, x, 2 k + 1] QPochhammer[ x, x^2, k + 1] ) , {k, 0, Sqrt @ n}], {x, 0, n}]]; (* _Michael Somos_, Jul 09 2015 *)

%t nmax = 100; CoefficientList[Series[Sum[x^(k^2+k)/(1-x)^2 * Product[(1+x^(2*j))/((1-x^(2*j))*(1-x^(2*j+1))^2), {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 11 2019 *)

%Y Cf. A006304, A006306.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_

%E Corrected and extended by _Dean Hickerson_, Dec 13 1999