A052860 -id:A052860 - cp's OEIS frontend

A007840 Number of factorizations of permutations of n letters into ordered cycles.

1, 1, 3, 14, 88, 694, 6578, 72792, 920904, 13109088, 207360912, 3608233056, 68495486640, 1408631978064, 31197601660080, 740303842925184, 18738231641600256, 503937595069600896, 14349899305396086912, 431322634732516137216, 13646841876634025159424

Offset: 0

Arnold Knopfmacher

nonn

a(n) is the number of ways to seat n people at an unspecified number of circular tables and then linearly order the nonempty tables. - Geoffrey Critzer, Mar 18 2009
The terms of this sequence for n >= 1 are the row sums of A008275^2, the unsigned version of A039814. - Peter Bala, Jul 22 2014
Signed sequence is the base for an Appell sequence of polynomials with the e.g.f. e^(x*t)/[log(1+t) + 1] = exp(P(.,x),t) that is the umbral compositional inverse for A238385, reverse of A111492, i.e., umbrally evaluated UP(n,P(.,t))= x^n = P(n,UP(.,t)) where UP(n,t) are the polynomials of A238385. Umbrally evaluated means letting (A(.,t))^n = A(n,t) after substituting A for the independent variable of the polynomial. - Tom Copeland, Nov 15 2014
a(n) is the number of unimodal rooted forests on n labeled nodes (i.e., those forests that avoid the patterns 213 and 312). - Kassie Archer, Aug 30 2018
Number of permutations of [n] where fixed points at index j are j-colored and all other points are unicolored. - Alois P. Heinz, Apr 24 2020

Alois P. Heinz, Table of n, a(n) for n = 0..420
K. Anders and K. Archer, Rooted forests that avoid sets of permutations, arXiv:1607.03046 [math.CO], 2016-2017.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 119.
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 122
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
A. Knopfmacher and J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
Chanchal Kumar and Amit Roy, Integer Sequences and Monomial Ideals, arXiv:2003.10098 [math.CO], 2020.

Cf. A052860. Row sums of unsigned version of A039814.

Cf. A238385, A111492.

a:= proc(n) a(n):= n!*`if`(n=0, 1, add(a(k)/(k!*(n-k)), k=0..n-1)) end:
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 06 2012

Table[Sum[Abs[StirlingS1[n, k]] k!, {k, 0, n}], {n, 0, 20}] (* Geoffrey Critzer, Mar 18 2009 *)

a(n)=n!*polcoeff(1/(1+log(1-x +x*O(x^n))),n) /* Paul D. Hanna, Jul 19 2006 */

{a(n)=local(CF=1+x*O(x)); for(k=0, n-1, CF=1/((n-k)-((n-k+1)\2)^2*x*CF)); n!*polcoeff(1/(1-x*CF), n)} /* Paul D. Hanna, Jul 19 2006 */

def A007840_list(len):
    f, R, C = 1, [1], [1]+[0]*len
    for n in (1..len):
        f *= n
        for k in range(n, 0, -1):
            C[k] = -C[k-1]*((k-1)/k if k>1 else 1)
        C[0] = sum((-1)^k*C[k] for k in (1..n))
        R.append(C[0]*f)
    return R
print(A007840_list(20)) # Peter Luschny, Feb 21 2016

a(n) = Sum_{k=1..n} k! * s(n, k), s(n, k) = unsigned Stirling number of first kind; E.g.f. 1/(1+log(1-z)).
For n>0, a(n) is the permanent of the n X n matrix with entries a(i, i) = i and a(i, j) = 1 elsewhere. - Philippe Deléham, Dec 09 2003
a(n) = A052860(n)/n for n >= 1.
a(n) = n!*Sum_{k=0..n-1} a(k)/k!/(n-k) for n >= 1 with a(0)=1. - Paul D. Hanna, Jul 19 2006
E.g.f.: B(A(x)) where B(x) = 1/(1-x) and A(x) = log(1/(1-x)). - Geoffrey Critzer, Mar 18 2009
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator exp(x)*d/dx. Cf. A006252. - Peter Bala, Nov 25 2011
E.g.f.: 1/(1+log(1-x)) = 1/(1 - x/(1 - x/(2 - x/(3 - 4*x/(4 - 4*x/(5 - 9*x/(6 - 9*x/(7 - 16*x/(8 - 16*x/(9 - ...)))))))))), a continued fraction. - Paul D. Hanna, Dec 31 2011
a(n) ~ n! * exp(n)/(exp(1)-1)^(n+1). - Vaclav Kotesovec, Jun 21 2013

Extended June 1995

A291978 Triangle read by rows, T(n, k) = (-1)^(n-k)n![t^k]([x^n] exp(x*t)/(1 + log(1+x))) for 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 14, 9, 3, 1, 88, 56, 18, 4, 1, 694, 440, 140, 30, 5, 1, 6578, 4164, 1320, 280, 45, 6, 1, 72792, 46046, 14574, 3080, 490, 63, 7, 1, 920904, 582336, 184184, 38864, 6160, 784, 84, 8, 1, 13109088, 8288136, 2620512, 552552, 87444, 11088, 1176, 108, 9, 1

Offset: 0

Peter Luschny, Sep 15 2017

			Triangle starts:
[1]
[1,           1]
[3,           2,      1]
[14,          9,      3,     1]
[88,         56,     18,     4,    1]
[694,       440,    140,    30,    5,   1]
[6578,     4164,   1320,   280,   45,   6,  1]
[72792,   46046,  14574,  3080,  490,  63,  7, 1]
[920904, 582336, 184184, 38864, 6160, 784, 84, 8, 1]

Row sums: A291978.
Columns: A007840 (c=1), A052860 (c=2).
Diagonal: A045943 (d=3).
Cf. A291980.

T_row := proc(n) exp(x*t)/(1 + log(1+x)): series(%, x, n+1):
seq(coeff((-1)^(n-k)*n!*coeff(%,x,n),t,k), k=0..n) end:
seq(T_row(n), n=0..9);

T[n_, k_] := Binomial[n, n - k]*Sum[j!*Abs[StirlingS1[n - k, j]], {j, 0, n - k}]; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* Detlef Meya, May 12 2024 *)

T(n, k) = binomial(n, n - k)*Sum_{j=0..n - k} j!*abs(Stirling1(n - k, j)). - Detlef Meya, May 12 2024

A355718 Expansion of e.g.f. exp( x/(1 + log(1-x)) ).

Original entry on oeis.org

1, 1, 3, 16, 117, 1071, 11725, 149122, 2158401, 35006941, 628552231, 12372376116, 264849067549, 6124239060915, 152099146415385, 4037206919213686, 114038575520545153, 3415098936831144537, 108065651366801837611, 3602585901321224507992

Offset: 0

Seiichi Manyama, Jul 15 2022

nonn

Cf. A355719, A355720.

Cf. A007840, A052860.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1+log(1-x)))))

a007840(n) = sum(k=0, n, k!*abs(stirling(n, k, 1)));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*a007840(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A052860(k) * binomial(n-1,k-1) * a(n-k).

a(n) ~ n^(n-1/4) * exp(1/4 - exp(-1) + 2*exp(-1/2)*sqrt(n)) / (sqrt(2) * (exp(1) - 1)^n). - Vaclav Kotesovec, Jul 15 2022

cp's OEIS Frontend

A007840 Number of factorizations of permutations of n letters into ordered cycles.

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A291978 Triangle read by rows, T(n, k) = (-1)^(n-k)n![t^k]([x^n] exp(x*t)/(1 + log(1+x))) for 0<=k<=n.

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Programs

Maple

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Formula

A355718 Expansion of e.g.f. exp( x/(1 + log(1-x)) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

cp's OEIS Frontend

A007840 Number of factorizations of permutations of n letters into ordered cycles.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A291978 Triangle read by rows, T(n, k) = (-1)^(n-k)*n!*[t^k]([x^n] exp(x*t)/(1 + log(1+x))) for 0<=k<=n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A355718 Expansion of e.g.f. exp( x/(1 + log(1-x)) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A291978 Triangle read by rows, T(n, k) = (-1)^(n-k)n![t^k]([x^n] exp(x*t)/(1 + log(1+x))) for 0<=k<=n.