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A216794 Number of set partitions of {1,2,...,n} with labeled blocks and a (possibly empty) subset of designated elements in each block. 2
1, 2, 12, 104, 1200, 17312, 299712, 6053504, 139733760, 3628677632, 104701504512, 3323151509504, 115063060869120, 4316023589937152, 174347763227738112, 7545919601962287104, 348366745238330081280, 17087957176042900815872, 887497598764802460352512 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

Eric Weisstein's MathWorld, 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

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

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. A006153, A055882, A080253.

Sequence in context: A320899 A194951 A104533 * A218300 A125031 A258230

Adjacent sequences:  A216791 A216792 A216793 * A216795 A216796 A216797

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Sep 16 2012

STATUS

approved

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Last modified February 18 02:15 EST 2019. Contains 320237 sequences. (Running on oeis4.)