A217389
Partial sums of the ordered Bell numbers (number of preferential arrangements) A000670.
Original entry on oeis.org
1, 2, 5, 18, 93, 634, 5317, 52610, 598445, 7685706, 109933269, 1732565842, 29824133437, 556682481818, 11198025452261, 241481216430114, 5557135898411469, 135927902927547370, 3521462566184392693, 96323049885512803826, 2774010846129897006941, 83898835844633970888762
Offset: 0
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A000670:=func;
[&+[A000670(k): k in [0..n]]: n in [0..19]]; // Bruno Berselli, Oct 03 2012
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b:= proc(n, k) option remember;
`if`(n=0, k!, k*b(n-1, k)+b(n-1, k+1))
end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n, 0)) end:
seq(a(n), n=0..23); # Alois P. Heinz, Feb 20 2025
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t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[t[k], {k, 0, n}], {n, 0, 100}]
(* second program: *)
Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Fubini[0, 1] = 1; Table[Fubini[n, 1], {n, 0, 20}] // Accumulate (* Jean-François Alcover, Mar 31 2016 *)
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t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum(t(k),k,0,n),n,0,40);
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for(n=0,30, print1(sum(k=0,n, sum(j=0,k, j!*stirling(k,j,2))), ", ")) \\ G. C. Greubel, Feb 07 2018
A217388
Alternating sums of the ordered Bell numbers (number of preferential arrangements) A000670.
Original entry on oeis.org
1, 0, 3, 10, 65, 476, 4207, 43086, 502749, 6584512, 95663051, 1526969522, 26564598073, 500293750308, 10141049220135, 220142141757718, 5095512540223637, 125275254488912264, 3260259408767933059, 89541327910560478074, 2588146468333823725041
Offset: 0
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List([0..30],n->Sum([0..n],k->(-1)^(n-k)*Sum([0..k], j-> Factorial(j)*Stirling2(k,j)))); # Muniru A Asiru, Feb 07 2018
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A000670:=func;
[&+[(-1)^(n-k)*A000670(k): k in [0..n]]: n in [0..20]]; // Bruno Berselli, Oct 03 2012
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with(combinat):
seq(sum((-1)^(n-k)*sum(factorial(j)*stirling2(k,j), j=0..k), k=0..n), n=0..30); # Muniru A Asiru, Feb 07 2018
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t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[(-1)^(n - k)t[k], {k, 0, n}], {n, 0, 100}]
(* second program: *)
Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Fubini[0, 1] = 1; a[n_] := Sum[(-1)^(n-k) Fubini[k, 1], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 31 2016 *)
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t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum((-1)^(n-k)*t(k),k,0,n),n,0,40);
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for(n=0,30, print1(sum(k=0,n, (-1)^(n-k)*sum(j=0,k, j!*stirling(k,j,2))), ", ")) \\ G. C. Greubel, Feb 07 2018
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a(n) = sum(k=0, n, k!*stirling(n+2,k+2,2)*(2^(k+1)-1)*(-1)^(n-k)) \\ Mikhail Kurkov, Aug 08 2025
A217391
Partial sums of the squares of the ordered Bell numbers (number of preferential arrangements) A000670.
Original entry on oeis.org
1, 2, 11, 180, 5805, 298486, 22228975, 2258856824, 300194704049, 50529463186170, 10505093602625139, 2643441560563225468, 791779611505017309493, 278371498870260182630654, 113516551713466910954246903, 53143864598655784249290736512, 28309328562668956145157858372537
Offset: 0
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A000670:=func;
[&+[A000670(k)^2: k in [0..n]]: n in [0..14]]; // Bruno Berselli, Oct 03 2012
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t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[t[k]^2, {k, 0, n}], {n, 0, 100}]
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t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum(t(k)^2,k,0,n),n,0,40);
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for(n=0,30, print1(sum(k=0,n, (sum(j=0,k, j!*stirling(k,j,2)))^2), ", ")) \\ G. C. Greubel, Feb 07 2018
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