A219779 1 = Sum_{n>=0} a(n) * x^n * (1 - (n+1)*x)^3.

1, 3, 15, 100, 819, 7890, 86995, 1077150, 14767575, 221751010, 3615510375, 63552101550, 1197196399675, 24048396026850, 512872089970875, 11569027674987550, 275113466366738175, 6876716833964911650, 180206071050892236175, 4939209747546747991950, 141294422480371952482275

Offset

0,2

Comments

Compare to: 1 = Sum_{n>=0} n! * x^n * (1 - (n+1)*x).

Compare to: 1 = Sum_{n>=0} A002720(n) * x^n * (1 - (n+1)*x)^2, where A002720(n) is the number of partial permutations of an n-set.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..439

Formula

a(n) = 3*n*a(n-1) - 3*(n-1)^2*a(n-2) + (n-2)^3*a(n-3). - Seiichi Manyama, Apr 27 2026

a(n) ~ c * n^(n - 1/12) * exp(2*sqrt(3*n) - n) * (1 + 79/(48*sqrt(3*n))), where c = 0.1290550041841... - Vaclav Kotesovec, May 10 2026

Example

E.g.f.: A(x) = 1 + 3*x + 15*x^2/2! + 100*x^3/3! + 819*x^4/4! + 7890*x^5/5! +...
By definition, the terms satisfy:
1 = (1-x)^3 + 3*x*(1-2*x)^3 + 15*x^2*(1-3*x)^3 + 100*x^3*(1-4*x)^3 + 819*x^4*(1-5*x)^3 + 7890*x^5*(1-6*x)^3 + 86995*x^6*(1-7*x)^3 +...

Mathematica

RecurrenceTable[{-(-2 + n)^3*a[-3 + n] + 3*(-1 + n)^2*a[-2 + n] - 3*n*a[-1 + n] + a[n] == 0, a[0] == 1, a[1] == 3, a[2] == 15}, a, {n, 0, 20}] (* Vaclav Kotesovec, May 09 2026 *)
(* Alternative: *)
nmax = 20; Round[Assuming[{x > 0}, CoefficientList[Series[E^(3*x/(1 - x)) * ((HypergeometricU[2/3, 1, 3/(-1 + x)] * (7*LaguerreL[-2/3, -3] - LaguerreL[1/3, -3]) - (3*HypergeometricU[-1/3, 1, -3] + 7*HypergeometricU[2/3, 1, -3]) * LaguerreL[-2/3, 3/(-1 + x)]) / ((-1 + x)*(3*HypergeometricU[-1/3, 1, -3] * LaguerreL[-2/3, -3] + HypergeometricU[2/3, 1, -3] * LaguerreL[1/3, -3]))), {x, 0, nmax}], x]] * Range[0, nmax]!] (* Vaclav Kotesovec, May 09 2026 *)

Prog

(PARI) {a(n)=polcoeff(x-sum(m=1, n-1, a(m)*x^m*(1-m*x+x*O(x^n))^3), n)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A000142, A002720, A219780.

Sequence in context: A381906 A147664 A384680 * A303651 A350954 A196962

Adjacent sequences: A219776 A219777 A219778 * A219780 A219781 A219782

Keyword

nonn,changed

Author

Paul D. Hanna, Nov 27 2012

Status

approved