OFFSET
0,3
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
FORMULA
E.g.f.: exp(-x)*((exp(x)-1)*exp(exp(x)-1)+1).
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
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
(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
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Sep 23 2003
STATUS
approved