A066398 Reversion of g.f. (with constant term included) for partition numbers.

1, -1, 0, 2, -3, 0, 5, 0, -21, 14, 117, -342, 210, 935, -2565, 1864, 2751, -3945, -8074, 4046, 108927, -333832, 246895, 887040, -2764795, 3062749, -1372098, 4775900, -9367698, -55130625, 299939766, -537241936, -140898285, 2464380030, -4060507784, 193070394

Offset: 0

Author

N. J. A. Sloane, Dec 25 2001

Keywords

• sign

Comments

See A301624 for the corresponding series reversion for the plane partition numbers A000219. - Peter Bala, Feb 09 2020

Links

• P. Bala, Representing a sequence as [x^n] G(x)^n
• Index entries for reversions of series

Crossrefs

Cf. A000041, A000203, A007312, A301624.

Programs

Maple

with(numtheory):
Order := 36:
Gser := solve(series(x*exp(add(sigma[1](n)*x^n/n, n = 1..35)), x) = y, x):
seq(coeff(Gser, y^k), k = 1..35); # Peter Bala, Feb 09 2020

Mathematica

nmax = 34; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - Product[ 1 - x^k*A[x]^k, {k, 1, n}] + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)

Formula

The o.g.f. A(x) = 1 - x + 2*x^3 - 3*x^4 + 5*x^6 - ... satisfies [x^n](1/A(x))^n = sigma(n) = A000203(n) for n >= 1. - Peter Bala, Aug 23 2015

G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k). - Ilya Gutkovskiy, Mar 21 2018