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A077607 Convolutory inverse of the factorial sequence. 4
1, -2, -2, -8, -44, -296, -2312, -20384, -199376, -2138336, -24936416, -314142848, -4252773824, -61594847360, -950757812864, -15586971531776, -270569513970944, -4959071121374720, -95721139472072192, -1941212789888952320, -41271304403571227648 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

|a(n)| is the number of permutations on [n] for which no proper initial interval of [n] is mapped to an interval. - David Callan, Nov 11 2003

LINKS

Table of n, a(n) for n=1..21.

Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

FORMULA

a(n) = -n!*a(1)-(n-1)!*a(2)-...-2!*a(n-1), with a(1)=1.

G.f.: 1/Sum_{k>=0} (k+1)!*x^k. - Vladeta Jovovic, May 04 2003

G.f. U(0)-x where U(k)= 1-x*(k+1)/(1-x*(k+2)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 15 2012

G.f.: A(x) = G(0)-x where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 26 2012.

G.f.: G(0)  where G(k) = 1 - x*(k+2)/( 1 - x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013

G.f.: (x - x^(2/3))/(Q(0) - 1), where Q(k) = 1 - (k+1)*x^(2/3)/(1 - x^(1/3)/(x^(1/3) - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013

G.f.: 1 - x - x/Q(0), where Q(k)= 1 + k*x - x*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 04 2013

G.f.: 2/G(0), where G(k)= 1 + 1/(1 - x*(k+2)/(x*(k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013

G.f.: 1/W(0), where W(k) = 1 - x*(k+2)/( x*(k+2) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 25 2013

G.f.: x/(1- Q(0)) - x, where Q(k) = 1  - (k+1)*x/(1  - (k+1)*x/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 18 2013

G.f.:1-x - x*T(0), where T(k) = 1 - x*(k+2)/( x*(k+2) - (1+k*x)*(1+x+k*x)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 07 2013

a(n) ~ -n! * (1 - 4/n - 8/n^3 - 76/n^4 - 752/n^5 - 8460/n^6 - 107520/n^7 - 1522124/n^8 - 23717424/n^9 - 402941324/n^10), for coefficients see A260491. - Vaclav Kotesovec, Jul 27 2015

a(n) = -2*A111529(n-2), for n>=2. - Vaclav Kotesovec, Jul 29 2015

EXAMPLE

a(4)= -8 = -24*1-6*(-2)-2*(-2). (a(1),a(2),...,a(n))(*)(1,2,3!,...,n!)=(1,0,0,...,0), where (*) denotes convolution.

MATHEMATICA

Clear[a]; a[1]=1; a[n_]:=a[n]=Sum[-(n-j+1)!*a[j], {j, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jul 27 2015 *)

PROG

(Sage)

def A077607_list(len):

    R, C = [1], [1]+[0]*(len-1)

    for n in (1..len-1):

        for k in range(n, 0, -1):

            C[k] = C[k-1] * (k+1)

        C[0] = -sum(C[k] for k in (1..n))

        R.append(C[0])

    return R

print A077607_list(21) # Peter Luschny, Feb 28 2016

CROSSREFS

Cf. A000142, A003319, A111529, A260491.

Sequence in context: A212307 A111605 A009544 * A264835 A032030 A184347

Adjacent sequences:  A077604 A077605 A077606 * A077608 A077609 A077610

KEYWORD

sign

AUTHOR

Clark Kimberling, Nov 11 2002

EXTENSIONS

More terms from Vaclav Kotesovec, Jul 29 2015

STATUS

approved

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Last modified March 28 19:06 EDT 2017. Contains 284246 sequences.