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 A077607 Convolutory inverse of the factorial sequence. 9
 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 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

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Last modified September 24 20:09 EDT 2018. Contains 315356 sequences. (Running on oeis4.)