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A087650 Sum_{k=0..n} (-1)^(n-k)*Bell(k). 4
1, 0, 2, 3, 12, 40, 163, 714, 3426, 17721, 98254, 580316, 3633281, 24011156, 166888166, 1216070379, 9264071768, 73600798036, 608476008123, 5224266196934, 46499892038438, 428369924118313, 4078345814329010, 40073660040755336 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) = number of set partitions of [n] that contain exactly one singleton block and all other blocks contain an entry > this singleton. For example, a(3)=3 counts 124/3, 134/2, 1/234 but not 123/4. - David Callan, Aug 27 2014

LINKS

Table of n, a(n) for n=0..23.

FORMULA

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

a(n) = (-1)^n + Bell(n) - A000296(n), with Bell(n)=A000110(n). - Wolfdieter Lang, Dec 01 2003

a(n) = A000296(n+1) + (-1)^n. - David Callan, Aug 27 2014

G.f.: 1/(1+x)/W(0), where W(k) = 1 - x/(1 - x*(k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 10 2014

EXAMPLE

G.f. = 1 + 2*x^2 + 3*x^3 + 12*x^4 + 40*x^5 + 163*x^6 + 714*x^7 + ...

MATHEMATICA

f[n_] := Sum[ StirlingS2[n, k], {k, 1, n}]; Table[(-1)^n + Sum[(-1)^(n - k)*f[k], {k, 0, n}], {n, 0, 23}] (* Robert G. Wilson v *)

Needs["DiscreteMath`Combinatorica`"]; Table[ Sum[(-1)^(n - k)*BellB[k], {k, 0, n}], {n, 0, 23}] (* Robert G. Wilson v *)

PROG

(Maxima) makelist(sum((-1)^(n-k)*belln(k), k, 0, n), n, 0, 40); // Emanuele Munarini, Sep 27 2012

(Sage)

def A087650_list(len): # After the formula of David Callan.

    if len == 1: return [1]

    if len == 2: return [1, 0]

    R = []; A = [1]; p = -1

    for i in (0..len-1):

        A.append(A[0] - A[i])

        A[i] = A[0]

        for k in range(i, 0, -1):

            A[k-1] += A[k]

        p = -p

        R.append(A[i+1] + p)

    return R

A087650_list(24) # Peter Luschny, Aug 28 2014

CROSSREFS

Cf. A000110, A000296, A005001, A005493.

Sequence in context: A012310 A082526 A151368 * A177699 A012514 A012511

Adjacent sequences:  A087647 A087648 A087649 * A087651 A087652 A087653

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Sep 23 2003

STATUS

approved

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Last modified December 19 16:42 EST 2014. Contains 252236 sequences.