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A051296 INVERT transform of factorial numbers.

%I A051296 #100 Apr 15 2024 13:12:49
%S A051296 1,1,3,11,47,231,1303,8431,62391,524495,4960775,52223775,605595319,
%T A051296 7664578639,105046841127,1548880173119,24434511267863,410503693136559,
%U A051296 7315133279097607,137787834979031839,2734998201208351479,57053644562104430735,1247772806059088954855
%N A051296 INVERT transform of factorial numbers.
%C A051296 a(n) = Sum[ a1!a2!...ak! ] where (a1,a2,...,ak) ranges over all compositions of n. a(n) = number of trees on [0,n] rooted at 0, consisting entirely of filaments and such that the non-root labels on each filament, when arranged in order, form an interval of integers. A filament is a maximal path (directed away from the root) whose interior vertices all have outdegree 1 and which terminates at a leaf. For example with n=3, a(n) = 11 counts all n^(n-2) = 16 trees on [0,3] except the 3 trees {0->1, 1->2, 1->3}, {0->2, 2->1, 2->3}, {0->3, 3->1, 3->2} (they fail the all-filaments test) and the 2 trees {0->2, 0->3, 3->1}, {0->2, 0->1, 1->3} (they fail the interval-of-integers test). - _David Callan_, Oct 24 2004
%C A051296 a(n) is the number of lists of "unlabeled" permutations whose total length is n. "Unlabeled" means each permutation is on an initial segment of the positive integers (cf. A090238). Example: with dashes separating permutations, a(3) = 11 counts 123, 132, 213, 231, 312, 321, 1-12, 1-21, 12-1, 21-1, 1-1-1. - _David Callan_, Sep 20 2007
%C A051296 Number of compositions of n where there are k! sorts of part k. - _Joerg Arndt_, Aug 04 2014
%D A051296 L. Comtet, Advanced Combinatorics, Reidel, 1974.
%H A051296 Alois P. Heinz and Vaclav Kotesovec, <a href="/A051296/b051296.txt">Table of n, a(n) for n = 0..440</a> (first 200 terms from Alois P. Heinz)
%H A051296 Jean-Paul Bultel, Ali Chouria, Jean-Gabriel Luque, and Olivier Mallet, <a href="http://hal.archives-ouvertes.fr/hal-00793788">Word symmetric functions and the Redfield-Polya theorem</a>, hal-00793788, 2013.
%H A051296 Louis Comtet, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5619085m/f9.image">Sur les coefficients de l'inverse de la série formelle Sum n! t^n</a>, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572.
%H A051296 Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, <a href="https://arxiv.org/abs/2310.06288">Catalan-Spitzer permutations</a>, arXiv:2310.06288 [math.CO], 2023. See p. 20.
%H A051296 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 A051296 Jun Yan, <a href="https://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv:2404.07958 [math.CO], 2024. See p. 7.
%F A051296 G.f.: 1/(1-Sum_{n>=1} n!*x^n).
%F A051296 a(0) = 1; a(n) = Sum_{k=1..n} a(n-k)*k! for n>0.
%F A051296 a(n) = Sum_{k>=0} A090238(n, k). - _Philippe Deléham_, Feb 05 2004
%F A051296 From _Gary W. Adamson_, Sep 26 2011: (Start)
%F A051296 a(n) is the upper left term of M^n, M = an infinite square production matrix as follows:
%F A051296 1, 1, 0, 0, 0, 0, ...
%F A051296 2, 0, 2, 0, 0, 0, ...
%F A051296 3, 0, 0, 3, 0, 0, ...
%F A051296 4, 0, 0, 0, 4, 0, ...
%F A051296 5, 0, 0, 0, 0, 5, ...
%F A051296 ... (End)
%F A051296 G.f.: 1 + x/(G(0) - 2*x) where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Dec 26 2012
%F A051296 a(n) ~ n! * (1 + 2/n + 7/n^2 + 35/n^3 + 216/n^4 + 1575/n^5 + 13243/n^6 + 126508/n^7 + 1359437/n^8 + 16312915/n^9 + 217277446/n^10), for coefficients see A260530. - _Vaclav Kotesovec_, Jul 28 2015
%F A051296 From _Peter Bala_, May 26 2017: (Start)
%F A051296 G.f. as an S-fraction: A(x) = 1/(1 - x/(1 - 2*x/(1 - x/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 3*x/(1 - n*x/(1 - (n - 1)*x/(1 - ...)))))))))). Cf. S-fraction for the o.g.f. of A000142.
%F A051296 A(x) = 1/(1 - x/(1 - x - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - 4*x/(1 - 4*x/(1 - ... ))))))))). (End)
%e A051296 a(4) = 47 = 1*24 + 1*6 + 3*2 + 11*1.
%e A051296 a(4) = 47, the upper left term of M^4.
%p A051296 a:= proc(n) option remember; `if`(n<1, 1,
%p A051296       add(a(n-i)*factorial(i), i=1..n))
%p A051296     end:
%p A051296 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 28 2015
%t A051296 CoefficientList[Series[Sum[Sum[k!*x^k, {k, 1, 20}]^n, {n, 0, 20}], {x, 0, 20}], x] (* _Geoffrey Critzer_, Mar 22 2009 *)
%o A051296 (Sage)
%o A051296 h = lambda x: 1/(1-x*hypergeometric((1, 2), (), x))
%o A051296 taylor(h(x),x,0,22).list() # _Peter Luschny_, Jul 28 2015
%o A051296 (Sage)
%o A051296 def A051296_list(len):
%o A051296     R, C = [1], [1]+[0]*(len-1)
%o A051296     for n in (1..len-1):
%o A051296         for k in range(n, 0, -1):
%o A051296             C[k] = C[k-1] * k
%o A051296         C[0] = sum(C[k] for k in (1..n))
%o A051296         R.append(C[0])
%o A051296     return R
%o A051296 print(A051296_list(23)) # _Peter Luschny_, Feb 21 2016
%Y A051296 Cf. A051295, row sums of A090238.
%Y A051296 Cf. A000142, A292778.
%K A051296 easy,nonn
%O A051296 0,3
%A A051296 _Leroy Quet_
%E A051296 Entry revised by _David Callan_, Sep 20 2007