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%I #56 Jan 11 2025 17:06:50
%S 1,0,2,3,12,40,163,714,3426,17721,98254,580316,3633281,24011156,
%T 166888166,1216070379,9264071768,73600798036,608476008123,
%U 5224266196934,46499892038438,428369924118313,4078345814329010,40073660040755336
%N a(n) = Sum_{k=0..n} (-1)^(n-k)*Bell(k).
%C 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
%C Partial sums are A173109. - _Vladimir Reshetnikov_, Oct 29 2015
%F E.g.f.: exp(-x)*((exp(x)-1)*exp(exp(x)-1)+1).
%F a(n) = (-1)^n + Bell(n) - A000296(n), with Bell(n) = A000110(n). - _Wolfdieter Lang_, Dec 01 2003
%F a(n) = A000296(n+1) + (-1)^n. - _David Callan_, Aug 27 2014
%F 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
%F a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n,k) * a(k-1). - _Ilya Gutkovskiy_, Mar 04 2021
%e G.f. = 1 + 2*x^2 + 3*x^3 + 12*x^4 + 40*x^5 + 163*x^6 + 714*x^7 + ...
%t 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_ *)
%t Needs["DiscreteMath`Combinatorica`"]; Table[ Sum[(-1)^(n - k)*BellB[k], {k, 0, n}], {n, 0, 23}] (* _Robert G. Wilson v_ *)
%o (Maxima) makelist(sum((-1)^(n-k)*belln(k),k,0,n),n,0,40); /* _Emanuele Munarini_, Sep 27 2012 */
%o (Sage)
%o def A087650_list(len): # After the formula of David Callan.
%o if len == 1: return [1]
%o if len == 2: return [1,0]
%o R = []; A = [1]; p = -1
%o for i in (0..len-1):
%o A.append(A[0] - A[i])
%o A[i] = A[0]
%o for k in range(i, 0, -1):
%o A[k-1] += A[k]
%o p = -p
%o R.append(A[i+1] + p)
%o return R
%o A087650_list(24) # _Peter Luschny_, Aug 28 2014
%o (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
%Y Cf. A000110, A000296, A005001, A005493, A173109.
%K nonn,changed
%O 0,3
%A _Vladeta Jovovic_, Sep 23 2003