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A051296
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INVERT transform of factorial numbers.
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15
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1, 1, 3, 11, 47, 231, 1303, 8431, 62391, 524495, 4960775, 52223775, 605595319, 7664578639, 105046841127, 1548880173119, 24434511267863, 410503693136559, 7315133279097607, 137787834979031839, 2734998201208351479, 57053644562104430735, 1247772806059088954855
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OFFSET
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0,3
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COMMENTS
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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
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
Number of compositions of n where there are k! sorts of part k. - Joerg Arndt, Aug 04 2014
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974.
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LINKS
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FORMULA
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G.f.: 1/(1-Sum_{n>=1} n!*x^n).
a(0) = 1; a(n) = Sum_{k=1..n} a(n-k)*k! for n>0.
a(n) is the upper left term of M^n, M = an infinite square production matrix as follows:
1, 1, 0, 0, 0, 0, ...
2, 0, 2, 0, 0, 0, ...
3, 0, 0, 3, 0, 0, ...
4, 0, 0, 0, 4, 0, ...
5, 0, 0, 0, 0, 5, ...
... (End)
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
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
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.
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)
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EXAMPLE
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a(4) = 47 = 1*24 + 1*6 + 3*2 + 11*1.
a(4) = 47, the upper left term of M^4.
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MAPLE
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a:= proc(n) option remember; `if`(n<1, 1,
add(a(n-i)*factorial(i), i=1..n))
end:
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MATHEMATICA
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CoefficientList[Series[Sum[Sum[k!*x^k, {k, 1, 20}]^n, {n, 0, 20}], {x, 0, 20}], x] (* Geoffrey Critzer, Mar 22 2009 *)
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PROG
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(Sage)
h = lambda x: 1/(1-x*hypergeometric((1, 2), (), x))
(Sage)
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
C[0] = sum(C[k] for k in (1..n))
R.append(C[0])
return R
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CROSSREFS
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KEYWORD
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easy,nonn,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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