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

Author

Clark Kimberling, Nov 11 2002

Keywords

• sign

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

Examples

			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.
		

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).

Crossrefs

Cf. A000142, A003319, A111529, A260491.

Programs

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 *)

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

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

Extensions

More terms from Vaclav Kotesovec, Jul 29 2015