A301624 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k)^k.

1, -1, -1, 4, 1, -17, -6, 118, -8, -876, 625, 5966, -7486, -41937, 75969, 306312, -768637, -2164992, 7487063, 14461466, -70259884, -89410774, 646971980, 459817892, -5861484630, -1128608133, 52082250637, -15894742662, -453574650852, 366848121166, 3866670213663, -5215687717614

Offset

0,4

Links

Table of n, a(n) for n=0..31.

Formula

From Peter Bala, Feb 09 2020: (Start)

A(x) = 1/x * series reversion of ( exp( Sum_{n >= 1} sigma_2(n)*x^n/n ) ), where sigma_2(n) = A001157(n).

Equivalently, the o.g.f. A(x) satisfies [x^n](1/A(x))^n = sigma_2(n) for n >= 1. Cf. A066398. (End)

A(x) equals (1/x) * series reversion of (x * the o.g.f. for the sequence of planar partitions A000219). - Peter Bala, Feb 11 2020

Example

G.f. A(x) = 1 - x - x^2 + 4*x^3 + x^4 - 17*x^5 - 6*x^6 + 118*x^7 - 8*x^8 - 876*x^9 + 625*x^10 + ...
G.f. A(x) satisfies: A(x) = (1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * (1 - x^4*A(x)^4)^4 * ...
log(A(x)) = -x - 3*x^2/2 + 8*x^3/3 + 13*x^4/4 - 51*x^5/5 - 120*x^6/6 + 538*x^7/7 + 781*x^8/8 - 5419*x^9/9 - 3053*x^10/10 + ... + A281267(n)*x^n/n + ...

Maple

with(numtheory):
Order := 33:
Gser := solve(series(x*exp(add(sigma[2](n)*x^n/n, n = 1..32)), x) = y, x):
seq(coeff(Gser, y^k), k = 1..32); # Peter Bala, Feb 09 2020

Crossrefs

Cf. A000219, A006195, A066398, A073592, A109085, A181315, A278428, A281267, A301455, A301456, A301625.

Sequence in context: A209411 A369912 A093035 * A126791 A052179 A171589

Adjacent sequences: A301621 A301622 A301623 * A301625 A301626 A301627

Keyword

sign

Author

Ilya Gutkovskiy, Mar 24 2018

Status

approved