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A077607 Convolutory inverse of the factorial sequence.

%I A077607 #85 Feb 29 2020 18:11:08
%S A077607 1,-2,-2,-8,-44,-296,-2312,-20384,-199376,-2138336,-24936416,
%T A077607 -314142848,-4252773824,-61594847360,-950757812864,-15586971531776,
%U A077607 -270569513970944,-4959071121374720,-95721139472072192,-1941212789888952320,-41271304403571227648
%N A077607 Convolutory inverse of the factorial sequence.
%C A077607 |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
%H A077607 Alois P. Heinz, <a href="/A077607/b077607.txt">Table of n, a(n) for n = 1..449</a>
%H A077607 Jean-Christophe Aval, Jean-Christophe Novelli, Jean-Yves Thibon, <a href="https://dmtcs.episciences.org/2892">The # product in combinatorial Hopf algebras</a>, 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).
%H A077607 Richard J. Martin, and Michael J. Kearney, <a href="http://dx.doi.org/10.1007/s00493-014-3183-3">Integral representation of certain combinatorial recurrences</a>, Combinatorica: 35:3 (2015), 309-315.
%H A077607 Ioannis Michos, Christina Savvidou, <a href="https://arxiv.org/abs/1803.08818">Enumeration of super-strong Wilf equivalence classes of permutations</a>, arXiv:1803.08818 [math.CO], 2018.
%H A077607 Vincent Pilaud, V. Pons, <a href="http://arxiv.org/abs/1606.09643">Permutrees</a>, arXiv preprint arXiv:1606.09643 [math.CO], 2016 (Unsigned version).
%F A077607 a(n) = -n!*a(1)-(n-1)!*a(2)-...-2!*a(n-1), with a(1)=1.
%F A077607 G.f.: 1/Sum_{k>=0} (k+1)!*x^k. - _Vladeta Jovovic_, May 04 2003
%F A077607 From _Sergei N. Gladkovskii_, Aug 15 2012 - Nov 07 2013: (Start)
%F A077607 Continued fractions:
%F A077607 G.f.: U(0) - x where U(k) = 1-x*(k+1)/(1-x*(k+2)/U(k+1)).
%F A077607 G.f.: A(x) = G(0) - x where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1).
%F A077607 G.f.: G(0) where G(k) = 1 - x*(k+2)/(1 - x*(k+1)/G(k+1)).
%F A077607 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))).
%F A077607 G.f.: 1 - x - x/Q(0), where Q(k)= 1 + k*x - x*(k+2)/Q(k+1).
%F A077607 G.f.: 2/G(0) where G(k)= 1 + 1/(1 - x*(k+2)/(x*(k+2) + 1/G(k+1))).
%F A077607 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)))).
%F A077607 G.f.: x/(1- Q(0)) - x, where Q(k) = 1  - (k+1)*x/(1  - (k+1)*x/Q(k+1)).
%F A077607 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)
%F A077607 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
%F A077607 a(n) = -2*A111529(n-2), for n>=2. - _Vaclav Kotesovec_, Jul 29 2015
%e A077607 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.
%p A077607 a:= proc(n) option remember; `if`(n=1, 1,
%p A077607       -add((n-i+1)!*a(i), i=1..n-1))
%p A077607     end:
%p A077607 seq(a(n), n=1..25);  # _Alois P. Heinz_, Dec 20 2017
%t A077607 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 *)
%t A077607 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_ *)
%o A077607 (Sage)
%o A077607 def A077607_list(len):
%o A077607     R, C = [1], [1]+[0]*(len-1)
%o A077607     for n in (1..len-1):
%o A077607         for k in range(n, 0, -1):
%o A077607             C[k] = C[k-1] * (k+1)
%o A077607         C[0] = -sum(C[k] for k in (1..n))
%o A077607         R.append(C[0])
%o A077607     return R
%o A077607 print(A077607_list(21)) # _Peter Luschny_, Feb 28 2016
%Y A077607 Cf. A000142, A003319, A111529, A260491.
%K A077607 sign
%O A077607 1,2
%A A077607 _Clark Kimberling_, Nov 11 2002
%E A077607 More terms from _Vaclav Kotesovec_, Jul 29 2015