A087650 a(n) = Sum_{k=0..n} (-1)^(n-k)*Bell(k).

1, 0, 2, 3, 12, 40, 163, 714, 3426, 17721, 98254, 580316, 3633281, 24011156, 166888166, 1216070379, 9264071768, 73600798036, 608476008123, 5224266196934, 46499892038438, 428369924118313, 4078345814329010, 40073660040755336

Offset: 0

Author

Vladeta Jovovic, Sep 23 2003

Keywords

• nonn

Comments

a(n) is the 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

Partial sums are A173109. - Vladimir Reshetnikov, Oct 29 2015

Examples

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

Crossrefs

Cf. A000110, A000296, A005001, A005493, A173109.

Programs

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 *)

Maxima

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

PARI

vector(30, n, n--; sum(k=0, n, (-1)^(n-k)*polcoeff(sum(i=0, k, prod( j=1, i, x / (1 - j*x)), x^k * O(x)), k))) \\ Altug Alkan, Oct 30 2015

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

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

a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n,k) * a(k-1). - Ilya Gutkovskiy, Mar 04 2021