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A087650
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a(n) = Sum_{k=0..n} (-1)^(n-k)*Bell(k).
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10
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1, 0, 2, 3, 12, 40, 163, 714, 3426, 17721, 98254, 580316, 3633281, 24011156, 166888166, 1216070379, 9264071768, 73600798036, 608476008123, 5224266196934, 46499892038438, 428369924118313, 4078345814329010, 40073660040755336
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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E.g.f.: exp(-x)*((exp(x)-1)*exp(exp(x)-1)+1).
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
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EXAMPLE
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G.f. = 1 + 2*x^2 + 3*x^3 + 12*x^4 + 40*x^5 + 163*x^6 + 714*x^7 + ...
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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