%I #100 Apr 15 2024 13:12:49

%S 1,1,3,11,47,231,1303,8431,62391,524495,4960775,52223775,605595319,

%T 7664578639,105046841127,1548880173119,24434511267863,410503693136559,

%U 7315133279097607,137787834979031839,2734998201208351479,57053644562104430735,1247772806059088954855

%N INVERT transform of factorial numbers.

%C 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 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 Number of compositions of n where there are k! sorts of part k. - _Joerg Arndt_, Aug 04 2014

%D L. Comtet, Advanced Combinatorics, Reidel, 1974.

%H 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 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 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 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 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 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 G.f.: 1/(1-Sum_{n>=1} n!*x^n).

%F a(0) = 1; a(n) = Sum_{k=1..n} a(n-k)*k! for n>0.

%F a(n) = Sum_{k>=0} A090238(n, k). - _Philippe Deléham_, Feb 05 2004

%F From _Gary W. Adamson_, Sep 26 2011: (Start)

%F a(n) is the upper left term of M^n, M = an infinite square production matrix as follows:

%F 1, 1, 0, 0, 0, 0, ...

%F 2, 0, 2, 0, 0, 0, ...

%F 3, 0, 0, 3, 0, 0, ...

%F 4, 0, 0, 0, 4, 0, ...

%F 5, 0, 0, 0, 0, 5, ...

%F ... (End)

%F 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 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 From _Peter Bala_, May 26 2017: (Start)

%F 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 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 a(4) = 47 = 1*24 + 1*6 + 3*2 + 11*1.

%e a(4) = 47, the upper left term of M^4.

%p a:= proc(n) option remember; `if`(n<1, 1,

%p add(a(n-i)*factorial(i), i=1..n))

%p end:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jul 28 2015

%t CoefficientList[Series[Sum[Sum[k!*x^k, {k, 1, 20}]^n, {n, 0, 20}], {x, 0, 20}], x] (* _Geoffrey Critzer_, Mar 22 2009 *)

%o (Sage)

%o h = lambda x: 1/(1-x*hypergeometric((1, 2), (), x))

%o taylor(h(x),x,0,22).list() # _Peter Luschny_, Jul 28 2015

%o (Sage)

%o def A051296_list(len):

%o R, C = [1], [1]+[0]*(len-1)

%o for n in (1..len-1):

%o for k in range(n, 0, -1):

%o C[k] = C[k-1] * k

%o C[0] = sum(C[k] for k in (1..n))

%o R.append(C[0])

%o return R

%o print(A051296_list(23)) # _Peter Luschny_, Feb 21 2016

%Y Cf. A051295, row sums of A090238.

%Y Cf. A000142, A292778.

%K easy,nonn

%O 0,3

%A _Leroy Quet_

%E Entry revised by _David Callan_, Sep 20 2007