A219780 1 = Sum_{n>=0} a(n) * x^n * (1 - (n+1)*x)^4.

1, 4, 26, 220, 2243, 26484, 353380, 5239276, 85243413, 1507394980, 28749072350, 587631913212, 12804803195383, 296121904536148, 7239552829750920, 186477285179206924, 5045665971430927721, 143034320139018008196, 4238027733918053839714, 130967841736577170487068

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..435

Formula

a(n) = 4*n*a(n-1) - 6*(n-1)^2*a(n-2) + 4*(n-2)^3*a(n-3) - (n-3)^4*a(n-4). - Seiichi Manyama, Apr 27 2026

a(n) ~ c * exp(2*sqrt((3 + sqrt(6))*n) - n) * n^(n - 1/4 - 1/(2*sqrt(6))) * (1 + (132 + 65*sqrt(6))/(48*sqrt(6*(3 + sqrt(6))*n))), where c = 0.0527883147221... - Vaclav Kotesovec, May 10 2026

Example

E.g.f.: A(x) = 1 + 4*x + 26*x^2/2! + 220*x^3/3! + 2243*x^4/4! + 26484*x^5/5! +...
By definition, the terms satisfy:
1 = (1-x)^4 + 4*x*(1-2*x)^4 + 26*x^2*(1-3*x)^4 + 220*x^3*(1-4*x)^4 + 2243*x^4*(1-5*x)^4 + 26484*x^5*(1-6*x)^4 + 353380*x^6*(1-7*x)^4 +...

Mathematica

RecurrenceTable[{(-3 + n)^4*a[-4 + n] - 4*(-2 + n)^3*a[-3 + n] + 6*(-1 + n)^2*a[-2 + n] - 4*n*a[-1 + n] + a[n] == 0, a[0] == 1, a[1] == 4, a[2] == 26, a[3] == 220}, a, {n, 0, 20}] (* Vaclav Kotesovec, May 09 2026 *)

Prog

(PARI) {a(n)=polcoeff(1-sum(m=0, n-1, a(m)*x^m*(1-(m+1)*x+x*O(x^n))^4), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A000142, A002720, A219779.

Sequence in context: A187826 A390741 A145347 * A259902 A381910 A089816

Adjacent sequences: A219777 A219778 A219779 * A219781 A219782 A219783

Keyword

nonn

Author

Paul D. Hanna, Nov 27 2012

Status

approved