A052882 A simple grammar: rooted ordered set partitions.

0, 1, 2, 9, 52, 375, 3246, 32781, 378344, 4912515, 70872610, 1124723193, 19471590876, 365190378735, 7376016877334, 159620144556645, 3684531055645648, 90366129593683035, 2346673806524446218, 64325158601880061137, 1856031746386568222660, 56231443721132068265415

Offset

0,3

Comments

Recurrence (see Mathematica line) is similar to that for Genocchi numbers A001469. - Wouter Meeussen, Jan 09 2001

Stirling transform of A024167(n) = [ 1, 1, 5, 14, 94, ...] is a(n) = [ 1, 2, 9, 52, 375, ...]. Stirling transform of a(n) = [ 0, 2, 9, 52, 375, ...] is A087301(n+1) = [ 0, 2, 3, 20, ...]. - Michael Somos, Mar 04 2004

Starting with offset 1 = the right border of triangle A208744. - Gary W. Adamson, Mar 05 2012

a(n) is the number of ordered set partitions of {1,2,...,n} such that the first block is a singleton. - Geoffrey Critzer, Jul 22 2013

Ramanujan gives a method of finding a continued fraction of the solution x of an equation 1 = x + a2*x^2 + ... and uses log(2) as the solution of 1 = x + x^2/2 + x^3/6 + ... as an example giving the sequence of simplified convergents as 0/1, 1/1, 2/3, 9/13, 52/75, 375/541, ... of which the sequence of numerators is this sequence while A000670 is the denominators. - Michael Somos, Jun 19 2015

References

S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 19.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Samuele Giraudo, Combinatorial operads from monoids, arXiv preprint arXiv:1306.6938 [math.CO], 2013.

Jia Huang and Erkko Lehtonen, Associative-commutative spectra for some varieties of groupoids, arXiv:2401.15786 [math.CO], 2024. See p. 14.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 855

Srinivasa Ramanujan, Notebook entry

Formula

E.g.f.: x / (2 - exp(x)).

a(n) = n * A000670(n-1) if n>0.

a(n) = (1/2)*sum(k=0, n-1, B_k*A000629(k)*binomial(n, k)) where B_k is the k-th Bernoulli number. - Benoit Cloitre, Oct 19 2005

a(n) ~ n!/(2*(log(2))^n). - Vaclav Kotesovec, Aug 09 2013

a(0) = 0, a(1) = 1; a(n) = n! * [x^n] exp(x)*Sum_{k=1..n-1} a(k)*x^k/k!. - Ilya Gutkovskiy, Oct 17 2017

Example

G.f. = x + 2*x^2 + 9*x^3 + 52*x^4 + 375*x^5 + 3246*x^6 + 32781*x^7 + ...

Maple

spec := [S, {C=Sequence(B), B=Set(Z, 1 <= card), S=Prod(Z, C)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
with(combinat): a:=n-> add(add(add((-1)^(k-i)*binomial(k, i)*i^(n-1), i=0..n-1), k=0..n-1), m=0..n-1): seq(a(n), n=0..20); # Zerinvary Lajos, Jun 03 2007
# next Maple program:
b:= proc(n, k) option remember;
     `if`(n<1, k!, k*b(n-1, k)+b(n-1, k+1))
    end:
a:= n-> b(n-1, 0)*n:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 15 2023

Mathematica

a[1] := 1; a[n_] := a[n]=Sum[ Binomial[n, m] a[ n-m], {m, 1, n-1}]
Range[0, 30]!* CoefficientList[Series[x/(2 - Exp[x]), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 06 2012 *)
a[ n_] := If[ n < 2, Boole[n == 1], n PolyLog[ 1 - n, 1/2] / 2]; (* Michael Somos, Jun 19 2015 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ x / (2 - Exp@x), {x, 0, n}]]; (* Michael Somos, Jun 19 2015 *)
Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Fubini[0, 1] = 1; a[n_] := n*Fubini[n-1, 1]; Table[ a[n], {n, 0, 18}] (* Jean-François Alcover, Mar 30 2016 *)

Prog

(PARI) {a(n) = if( n<0, 0, n! * polcoeff( subst( x / (1 - y), y, exp(x + x*O(x^n)) - 1), n))};
(Python)
from math import factorial
from sympy.functions.combinatorial.numbers import stirling
def A052882(n): return n*sum(factorial(k)*stirling(n-1, k) for k in range(n)) # Chai Wah Wu, Apr 15 2023

Crossrefs

Cf. A001469.

Cf. A000629, A000670, A024167, A087301.

Cf. A108744.

Sequence in context: A006152 A369551 A143508 * A248440 A330200 A143922

Adjacent sequences: A052879 A052880 A052881 * A052883 A052884 A052885

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Status

approved