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Binomial transform of A033312.
3

%I #31 Nov 23 2016 16:19:16

%S 0,0,1,8,49,294,1893,13572,109345,985898,9863077,108503064,1302057249,

%T 16926789294,236975148421,3554627439308,56874039487681,

%U 966858672273618,17403456103022277,330665665961879712,6613313319247031425,138879579704207582870

%N Binomial transform of A033312.

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

%H Vincenzo Librandi, <a href="/A097204/b097204.txt">Table of n, a(n) for n = 0..200</a>

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

%F 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

%F 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

%F a(n) ~ n! * exp(1). - _Vaclav Kotesovec_, Feb 09 2014

%F a(n) = exp(1)*Gamma(n+1, 1) - 2^n and a(n) ~ exp(1)*n! - 2^n. - _Peter Luschny_, Sep 02 2014

%F a(n) = A000522(n) - 2^n. - _Anton Zakharov_, Nov 20 2016

%F E.g.f.: (1/(1 - x) - exp(x))*exp(x). - _Ilya Gutkovskiy_, Nov 20 2016

%F 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

%e G.f. = x^2 + 8*x^3 + 49*x^4 + 294*x^5 + 1893*x^6 + 13572*x^7 + 109345*x^8 + ...

%e 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.

%p 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

%p a := n -> exp(1)*GAMMA(n+1, 1)-2^n;

%p seq(simplify(a(n)), n=0..20); # _Peter Luschny_, Sep 02 2014

%t Table[Sum[Binomial[n,k]*(k!-1),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Feb 09 2014 *)

%o (PARI) {a(n) = sum(k=0, n, binomial(n, k) * (k! - 1))}; /* _Michael Somos_, Nov 23 2016 */

%Y Cf. A033312, A000522.

%K nonn

%O 0,4

%A _Ross La Haye_, Sep 16 2004

%E More terms from _R. J. Mathar_, Aug 05 2007