Original entry on oeis.org
1, 1, 6, 65, 1125, 28132, 950649, 41475961, 2259756900, 149874308367, 11858161118925, 1101069785060610, 118366544943589215, 14564702419742606497, 2031425158227034739646, 318472106732688712103885, 55708816671530680003669185, 10803156636116962310987233404
Offset: 0
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a:= n-> combinat[bell](n)*add(Stirling2(n, k)*k!, k=0..n): seq(a(n), n=0..19); # Zerinvary Lajos, Sep 30 2006
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Table[BellB[n]*Sum[StirlingS2[n, k]*k!, {k, 0, n}], {n, 0, 17}] (* James C. McMahon, Oct 11 2024 *)
Original entry on oeis.org
1, 1, 7, 85, 1587, 41981, 1484643, 67306429, 3790883659, 258899180989, 21029065282803, 1999625128004813, 219691693064750283, 27580289062408474861, 3919060527556589637043, 625165018565884343909053
Offset: 0
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read "transforms" ; A000670 := proc(n) local k ; if n = 0 then 1; else add(k!*combinat[stirling2](n,k),k=1..n) ; fi ; end: A000110 := proc(n) local k ; add(combinat[stirling2](n,k),k=0..n) ; end: A104600 := proc(n) local k ; add(combinat[stirling1](n,k)*A000670(k)*A000110(k),k=0..n) ; end: A121020 := proc(nmax) local a104600 ; a104600 := [seq(A104600(n),n=0..nmax)] ; LAH(a104600) ; end: A121020(20) ; # R. J. Mathar, Jan 21 2008
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a[n_] := a[n] = (1/(2 E)) Sum[Sum[Product[r s + k, {k, 0, n - 1}]/(2^r s!), {r, 0, Infinity}], {s, 0, Infinity}];
Reap[For[n = 0, n <= 80, n++, Print[n, " ", a[n]]; Sow[a[n]]]][[2, 1]] (* Jean-François Alcover, Apr 04 2020 *)
A323128
Number T(n,k) of colored set partitions of [n] where elements of subsets have distinct colors and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 4, 0, 1, 18, 30, 0, 1, 74, 360, 360, 0, 1, 310, 3450, 8880, 6240, 0, 1, 1382, 31770, 160080, 271800, 146160, 0, 1, 6510, 298662, 2635920, 8152200, 10190880, 4420080, 0, 1, 32398, 2918244, 42687960, 214527600, 468669600, 460474560, 166924800
Offset: 0
T(3,2) = 18: 1a|2a3b, 1a|2b3a, 1b|2a3b, 1b|2b3a, 1a3b|2a, 1b3a|2a, 1a3b|2b, 1b3a|2b, 1a2b|3a, 1b2a|3a, 1a2b|3b, 1b2a|3b, 1a|2a|3b, 1a|2b|3a, 1b|2a|3a, 1a|2b|3b, 1b|2a|3b, 1b|2b|3a.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 4;
0, 1, 18, 30;
0, 1, 74, 360, 360;
0, 1, 310, 3450, 8880, 6240;
0, 1, 1382, 31770, 160080, 271800, 146160;
0, 1, 6510, 298662, 2635920, 8152200, 10190880, 4420080;
...
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A:= proc(n, k) option remember; `if`(n=0, 1, add(k!/(k-j)!
*binomial(n-1, j-1)*A(n-j, k), j=1..min(k, n)))
end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
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A[n_, k_] := A[n, k] = If[n==0, 1, Sum[k!/(k - j)! Binomial[n - 1, j - 1]* A[n - j, k], {j, Min[k, n]}]];
T[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)
Showing 1-3 of 3 results.