A006305 Taylor series related to one in Ramanujan's Lost Notebook. (Formerly M1014)

1, 2, 4, 6, 10, 16, 25, 38, 58, 84, 122, 174, 244, 338, 465, 630, 850, 1136, 1508, 1988, 2608, 3398, 4408, 5688, 7306, 9342, 11900, 15090, 19070, 24008, 30122, 37666, 46955, 58348, 72302, 89338, 110094, 135316, 165912, 202924, 247632, 301508

Offset

0,2

References

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

G. E. Andrews, Mordell integrals and Ramanujan's "Lost" Notebook, pp. 10-48 of Analytic Number Theory (Philadelphia 1980), Lect. Notes Math. 899 (1981).

Formula

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).

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

Example

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 + ...

Mathematica

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}]
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 *)
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 *)

Crossrefs

Cf. A006304, A006306.

Sequence in context: A211971 A305498 A028445 * A067247 A017985 A327474

Adjacent sequences: A006302 A006303 A006304 * A006306 A006307 A006308

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Corrected and extended by Dean Hickerson, Dec 13 1999

Status

approved