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

Author

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

Keywords

• easy
• nonn

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

Examples

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

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

Crossrefs

Cf. A001469.

Cf. A000629, A000670, A024167, A087301.

Cf. A108744.

Programs

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 *)

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

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