A022693 Expansion of Product_{m>=1} 1/(1 + m*q^m).

1, -1, -1, -2, 2, -1, 4, -1, 18, -22, 12, -26, 67, -86, 42, -235, 432, -364, 506, -868, 1434, -2396, 2225, -3348, 10842, -11822, 8049, -24468, 36662, -40024, 69766, -96052, 171976, -278242, 251886, -419723, 885806, -998468, 1103660, -2381042, 4009539, -4478416, 6372514, -9913690

Offset

0,4

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Formula

From Vaclav Kotesovec, Dec 15 2015: (Start)

a(n) ~ (-1)^n * c * 3^(n/3), where

c = 2.0319526534291644237634198503666896166412... if mod(n,3) = 0

c = 1.8420902462379331740718256785549611496880... if mod(n,3) = 1

c = 1.6677871810486313099783673373643842640151... if mod(n,3) = 2.

(End)

From Benedict W. J. Irwin, Mar 19 2017: (Start)

Conjecture: a(n) = Sum_{i_1,i_2,i_3,...}[(-1)^(i_1+i_2+i_3+...)*Product_{n>0} n^i_n], where the sum is over all valid sequences of positive i_k such that i_1+2*i_2+3*i_3+4*i_4+...= n.

Examples: Setting i_k=0 unless explicitly mentioned.

n=1, (i_1=1), a(1)= -1^1 = -1.

n=2, (i_1=2) or (i_2=1), a(2) = 1^2 - 2^1 = -1.

n=3, (i_1=3) or (i_1=1,i_2=1) or (i_3=1), a(3)=-1^3 + 1^1*2^1 - 3^1 = -2.

(End)

Mathematica

nmax = 40; CoefficientList[Series[Product[1/(1 + k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 15 2015 *)
nmax = 40; CoefficientList[Series[Exp[-Sum[(-1)^(j+1)*PolyLog[-j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 15 2015 *)

Prog

(PARI) m=50; q='q+O('q^m); Vec(prod(n=1, m, 1/(1+n*q^n))) \\ G. C. Greubel, Feb 25 2018
(Magma) Coefficients(&*[1/(1+m*x^m):m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018

Crossrefs

Cf. A006906, A022629, A022661, A266971.

Sequence in context: A134666 A129712 A051720 * A174498 A146478 A289500

Adjacent sequences: A022690 A022691 A022692 * A022694 A022695 A022696

Keyword

sign

Author

N. J. A. Sloane

Status

approved