A077607 Convolutory inverse of the factorial sequence.

1, -2, -2, -8, -44, -296, -2312, -20384, -199376, -2138336, -24936416, -314142848, -4252773824, -61594847360, -950757812864, -15586971531776, -270569513970944, -4959071121374720, -95721139472072192, -1941212789888952320, -41271304403571227648

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

Alois P. Heinz, Table of n, a(n) for n = 1..449

Jean-Christophe Aval, Jean-Christophe Novelli, Jean-Yves Thibon, The # product in combinatorial Hopf algebras, dmtcs:2892 - Discrete Mathematics & Theoretical Computer Science, January 1, 2011, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011).

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

Ioannis Michos, Christina Savvidou, Enumeration of super-strong Wilf equivalence classes of permutations, arXiv:1803.08818 [math.CO], 2018.

Vincent Pilaud, V. Pons, Permutrees, arXiv preprint arXiv:1606.09643 [math.CO], 2016 (Unsigned version).

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

From Sergei N. Gladkovskii, Aug 15 2012 - Nov 07 2013: (Start)

Continued fractions:

G.f.: U(0) - x where U(k) = 1-x*(k+1)/(1-x*(k+2)/U(k+1)).

G.f.: A(x) = G(0) - x where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1).

G.f.: G(0) where G(k) = 1 - x*(k+2)/(1 - x*(k+1)/G(k+1)).

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))).

G.f.: 1 - x - x/Q(0), where Q(k)= 1 + k*x - x*(k+2)/Q(k+1).

G.f.: 2/G(0) where G(k)= 1 + 1/(1 - x*(k+2)/(x*(k+2) + 1/G(k+1))).

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)))).

G.f.: x/(1- Q(0)) - x, where Q(k) = 1 - (k+1)*x/(1 - (k+1)*x/Q(k+1)).

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)). (End)

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.

Maple

a:= proc(n) option remember; `if`(n=1, 1,
      -add((n-i+1)!*a(i), i=1..n-1))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Dec 20 2017

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 *)
terms=21; 1/Sum[(k+1)!*x^k, {k, 0, terms}]+O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Dec 20 2017, after Vladeta Jovovic *)

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