A216794 Number of set partitions of {1,2,...,n} with labeled blocks and a (possibly empty) subset of designated elements in each block.

1, 2, 12, 104, 1200, 17312, 299712, 6053504, 139733760, 3628677632, 104701504512, 3323151509504, 115063060869120, 4316023589937152, 174347763227738112, 7545919601962287104, 348366745238330081280, 17087957176042900815872, 887497598764802460352512

Offset

0,2

Links

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

José A. Adell, Beáta Bényi, Venkat Murali, and Sithembele Nkonkobe, Generalized Barred Preferential Arrangements, Transactions on Combinatorics (2022).

Sithembele Nkonkobe, Venkat Murali, and Béata Bényi, Generalised Barred Preferential Arrangements, arXiv:1907.08944 [math.CO], 2019.

Eric Weisstein's World of Mathematics, Polylogarithm.

Formula

E.g.f.: 1/(2 - exp(2*x)).

E.g.f.: 1 + 2*x/(G(0) - 2*x) where G(k) = 2*k+1 - x*2*(2*k+1)/(2*x + (2*k+2)/(1 + 2*x/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 26 2012

E.g.f.: 1 + 2*x/( G(0) - 2*x ) where G(k) = 1 - 2*x/(1 + (1*k+1)/G(k+1)); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 02 2013

G.f.: 1/G(0) where G(k) = 1 - x*(2*k+2)/( 1 - 4*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013

a(n) ~ n! * (2/log(2))^n/log(4). - Vaclav Kotesovec, Sep 24 2013

G.f.: T(0)/(1-2*x), where T(k) = 1 - 8*x^2*(k+1)^2/( 8*x^2*(k+1)^2 - (1-2*x-6*x*k)*(1-8*x-6*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2013

From Vladimir Reshetnikov, Oct 31 2015: (Start)

a(n) = (-1)^(n+1)*(Li_{-n}(sqrt(2)) + Li_{-n}(-sqrt(2)))/4, where Li_n(x) is the polylogarithm.

Li_{-n}(sqrt(2)) = (-1)^(n+1)*(2*a(n) + A080253(n)*sqrt(2)).

(End)

a(n) = 2^(n-1)*(Li_{-n}(1/2) + 0^n) with 0^0=1. - Peter Luschny, Nov 03 2015

From Peter Bala, Oct 18 2023: (Start)

a(n) = 2^n * A000670(n)

Inverse binomial transform of A080253.

The sequence is the first column of the array (2*I - P^2)^(-1), where P denotes Pascal's triangle A007318. (End)

Maple

a := n -> 2^(n-1)*(polylog(-n, 1/2)+`if`(n=0, 1, 0)):
seq(round(evalf(a(n), 32)), n=0..18); # Peter Luschny, Nov 03 2015
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*2^j, j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 04 2019

Mathematica

nn=25; a=Exp[2x]-1; Range[0, nn]!CoefficientList[Series[1/(1-a), {x, 0, nn}], x]
Round@Table[(-1)^(n+1) (PolyLog[-n, Sqrt[2]] + PolyLog[-n, -Sqrt[2]])/4, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

Prog

(Sage)
def A216794(n):
    return 2^n*add(add((-1)^(j-i)*binomial(j, i)*i^n for i in range(n+1)) for j in range(n+1))
[A216794(n) for n in range(18)] # Peter Luschny, Jul 22 2014
(PARI) a(n) = 2^(n-1)*(polylog(-n, 1/2) + 0^n); \\ Michel Marcus, May 30 2018

Crossrefs

Cf. A000556, A000670, A006153, A055882, A080253.

Sequence in context: A320899 A194951 A104533 * A218300 A125031 A258230

Adjacent sequences: A216791 A216792 A216793 * A216795 A216796 A216797

Keyword

nonn,easy

Author

Geoffrey Critzer, Sep 16 2012

Status

approved