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

Author

Ross La Haye, Sep 16 2004

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A033312, A000522.

Programs

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 *)

PARI

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

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

Extensions

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