A292778 -id:A292778 - cp's OEIS frontend

A051296 INVERT transform of factorial numbers.

1, 1, 3, 11, 47, 231, 1303, 8431, 62391, 524495, 4960775, 52223775, 605595319, 7664578639, 105046841127, 1548880173119, 24434511267863, 410503693136559, 7315133279097607, 137787834979031839, 2734998201208351479, 57053644562104430735, 1247772806059088954855

Offset: 0

Leroy Quet

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

			a(4) = 47 = 1*24 + 1*6 + 3*2 + 11*1.
a(4) = 47, the upper left term of M^4.

L. Comtet, Advanced Combinatorics, Reidel, 1974.

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..440 (first 200 terms from Alois P. Heinz)
Jean-Paul Bultel, Ali Chouria, Jean-Gabriel Luque, and Olivier Mallet, Word symmetric functions and the Redfield-Polya theorem, hal-00793788, 2013.
Louis Comtet, Sur les coefficients de l'inverse de la série formelle Sum n! t^n, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572.
Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, Catalan-Spitzer permutations, arXiv:2310.06288 [math.CO], 2023. See p. 20.
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 7.

Cf. A051295, row sums of A090238.

Cf. A000142, A292778.

a:= proc(n) option remember; `if`(n<1, 1,
      add(a(n-i)*factorial(i), i=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 28 2015

CoefficientList[Series[Sum[Sum[k!*x^k, {k, 1, 20}]^n, {n, 0, 20}], {x, 0, 20}], x] (* Geoffrey Critzer, Mar 22 2009 *)

h = lambda x: 1/(1-x*hypergeometric((1, 2), (), x))
taylor(h(x),x,0,22).list() # Peter Luschny, Jul 28 2015

def A051296_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
        C[0] = sum(C[k] for k in (1..n))
        R.append(C[0])
    return R
print(A051296_list(23)) # Peter Luschny, Feb 21 2016

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) = Sum_{k>=0} A090238(n, k). - Philippe Deléham, Feb 05 2004
From Gary W. Adamson, Sep 26 2011: (Start)
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
From Peter Bala, May 26 2017: (Start)
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)

Entry revised by David Callan, Sep 20 2007

A307064 Expansion of 1 - 1/Sum_{k>=0} k!!*x^k.

Original entry on oeis.org

0, 1, 1, 0, 3, 1, 18, 13, 155, 168, 1691, 2381, 22022, 37401, 331087, 649036, 5626103, 12372161, 106486594, 257573405, 2220690451, 5824952232, 50593271507, 142387607469, 1250521775454, 3745193283657, 33338037080183, 105558942751948, 953776675614223

Offset: 0

Ilya Gutkovskiy, Mar 21 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..800

Cf. A000698, A003319, A006882, A292778.

m:=80;
F2:= func< n | n mod 2 eq 0 select Round(2^(n/2)*Gamma(n/2+1)) else Round( Gamma((n+3)/2)*Binomial(n+1, Floor((n+1)/2))/2^((n+1)/2) ) >;
R:=PowerSeriesRing(Rationals(), m);
[0] cat Coefficients(R!( 1 - 1/(&+[F2(j)*x^j : j in [0..m+2]]) )); // G. C. Greubel, Jan 24 2024

nmax = 28; CoefficientList[Series[1 - 1/Sum[k!! x^k, {k, 0, nmax}], {x, 0, nmax}], x]
a[0] = 0; a[n_]:= a[n] = n!! - Sum[k!! a[n-k], {k,n-1}];
Table[a[n], {n, 0, 28}]

from sympy import factorial2
m=80;
def f(x): return 1 - 1/sum(factorial2(k)*x^k for k in range(m+1))
def A307063_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307063_list(m) # G. C. Greubel, Jan 24 2024

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

A295553 Expansion of 1/(1 - Sum_{k>=1} (2k-1)!!x^k).

Original entry on oeis.org

1, 1, 4, 22, 154, 1330, 13882, 171802, 2474098, 40738594, 755322778, 15566915770, 352862768434, 8720662458754, 233285616212506, 6713983428179098, 206813607458357746, 6788092999359053410, 236481982146071359258, 8714521818620631672058, 338660320676350494328882, 13841377309645038610883266

Offset: 0

Ilya Gutkovskiy, Nov 23 2017

nonn

Invert transform of A001147.

Number of compositions (ordered partitions) of n where there are 1*3*5*...*(2*k-1) sorts of part k.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers
Index entries for sequences related to compositions

Cf. A001147, A051296, A185971, A292778.

nmax = 21; CoefficientList[Series[1/(1 - Sum[(2 k - 1)!! x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 21; CoefficientList[Series[1 + x/(1 - 2 x + ContinuedFractionK[-k x, 1, {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(2 k - 1)!! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

G.f.: 1/(1 - Sum_{k>=1} A001147(k)*x^k).
G.f.: 1 + x/(1 - 2*x - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - 6*x/(1 - ...)))))), a continued fraction.
a(0) = 1; a(n) = Sum_{k=1..n} (2*k-1)!!*a(n-k).

A305535 Expansion of 1/(1 - x/(1 - 2x/(1 - 2x/(1 - 4x/(1 - 4x/(1 - 6x/(1 - 6x/(1 - ...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 3, 13, 75, 557, 5179, 58589, 784715, 12154061, 213593563, 4195613373, 91031201643, 2160916171181, 55687501548539, 1547866851663261, 46150908197995403, 1469089501918434957, 49722765216242122267, 1782934051704982201469, 67514992620138056010667

Offset: 0

Ilya Gutkovskiy, Jun 04 2018

nonn

Invert transform of A000165, shifted right one place.

N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Cf. A000165, A051295, A051296, A112934, A141307, A292778.

nmax = 20; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[-2 Floor[(k + 1)/2] x, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 20; CoefficientList[Series[1/(1 - Sum[2^(k - 1) (k - 1)! x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[2^(k - 1) (k - 1)! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

a(n) ~ 2^(n-1) * (n-1)!. - Vaclav Kotesovec, Sep 18 2021

A335848 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k!! * a(n-k).

Original entry on oeis.org

1, 1, 4, 21, 152, 1355, 14568, 182427, 2612224, 42073209, 752981280, 14823367845, 318347145216, 7406554353939, 185573713100160, 4981725842622795, 142650055922872320, 4340032650657965745, 139809806502181765632, 4754045863586538697077, 170163141506896128122880

Offset: 0

Ilya Gutkovskiy, Jun 26 2020

nonn

Cf. A002866, A006882, A292778, A308939.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k!! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; CoefficientList[Series[1/(1 - x Exp[x^2/2] (1 + Sqrt[Pi/2] Erf[x/Sqrt[2]])), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: 1 / (1 - x * exp(x^2/2) * (1 + sqrt(Pi/2) * erf(x/sqrt(2)))), where erf() is the error function.

cp's OEIS Frontend

A051296 INVERT transform of factorial numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Sage

Formula

Extensions

A307064 Expansion of 1 - 1/Sum_{k>=0} k!!*x^k.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A295553 Expansion of 1/(1 - Sum_{k>=1} (2*k-1)!!*x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A305535 Expansion of 1/(1 - x/(1 - 2*x/(1 - 2*x/(1 - 4*x/(1 - 4*x/(1 - 6*x/(1 - 6*x/(1 - ...)))))))), a continued fraction.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A335848 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k!! * a(n-k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A295553 Expansion of 1/(1 - Sum_{k>=1} (2k-1)!!x^k).

A305535 Expansion of 1/(1 - x/(1 - 2x/(1 - 2x/(1 - 4x/(1 - 4x/(1 - 6x/(1 - 6x/(1 - ...)))))))), a continued fraction.