A097204 Binomial transform of A033312.

0, 0, 1, 8, 49, 294, 1893, 13572, 109345, 985898, 9863077, 108503064, 1302057249, 16926789294, 236975148421, 3554627439308, 56874039487681, 966858672273618, 17403456103022277, 330665665961879712, 6613313319247031425, 138879579704207582870

Offset

0,4

Comments

a(n) = Sum_{k=0...n} n!(k!-1)/(k!(n-k)!).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = Sum_{k=0...n}(P(n,k) - binomial(n,k)).

Conjecture: a(n) +(-n-5)*a(n-1) +(5*n+3)*a(n-2) +4*(-2*n+3)*a(n-3) +4*(n-3)*a(n-4)=0. - R. J. Mathar, Nov 16 2012

Recurrence: (n-2)*a(n) = (n^2 + n - 4)*a(n-1) - (n-1)*(3*n-2)*a(n-2) + 2*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Feb 09 2014

a(n) ~ n! * exp(1). - Vaclav Kotesovec, Feb 09 2014

a(n) = exp(1)*Gamma(n+1, 1) - 2^n and a(n) ~ exp(1)*n! - 2^n. - Peter Luschny, Sep 02 2014

a(n) = A000522(n) - 2^n. - Anton Zakharov, Nov 20 2016

E.g.f.: (1/(1 - x) - exp(x))*exp(x). - Ilya Gutkovskiy, Nov 20 2016

0 = a(n)*(+16*a(n+1) - 80*a(n+2) + 92*a(n+3) - 36*a(n+4) + 4*a(n+5)) + a(n+1)*(+16*a(n+1) + 12*a(n+2) - 96*a(n+3) + 56*a(n+4) - 8*a(n+5)) + a(n+2)*(+44*a(n+2) - 19*a(n+3) - 19*a(n+4) + 5*a(n+5)) + a(n+3)*(+17*a(n+3) - 4*a(n+4) - a(n+5)) + a(n+4)*(+a(n+4)) for n>=0

Example

G.f. = x^2 + 8*x^3 + 49*x^4 + 294*x^5 + 1893*x^6 + 13572*x^7 + 109345*x^8 + ...
a(2) = 1 because P(2,0) = 1, P(2,1) = 2, P(2,2) = 2 while C(2,0) = 1, C(2,1) = 2, C(2,2) = 1 and 1 - 1 + 2 - 2 + 2 - 1 = 1.

Maple

A033312 := proc(n) factorial(n)-1; end: A097204 := proc(n) add( binomial(n, k)*A033312(k), k=1..n) ; end: seq(A097204(n), n=0..30) ; # R. J. Mathar, Aug 05 2007
a := n -> exp(1)*GAMMA(n+1, 1)-2^n;
seq(simplify(a(n)), n=0..20); # Peter Luschny, Sep 02 2014

Mathematica

Table[Sum[Binomial[n, k]*(k!-1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 09 2014 *)

Prog

(PARI) {a(n) = sum(k=0, n, binomial(n, k) * (k! - 1))}; /* Michael Somos, Nov 23 2016 */

Crossrefs

Cf. A033312, A000522.

Sequence in context: A001108 A115598 A261668 * A037539 A037483 A188708

Adjacent sequences: A097201 A097202 A097203 * A097205 A097206 A097207

Keyword

nonn

Author

Ross La Haye, Sep 16 2004

Extensions

More terms from R. J. Mathar, Aug 05 2007

Status

approved