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 A097204 Binomial transform of A033312. 3
 0, 0, 1, 8, 49, 294, 1893, 13572, 109345, 985898, 9863077, 108503064, 1302057249, 16926789294, 236975148421, 3554627439308, 56874039487681, 966858672273618, 17403456103022277, 330665665961879712, 6613313319247031425, 138879579704207582870 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified June 3 04:09 EDT 2023. Contains 363103 sequences. (Running on oeis4.)