A122454
A triangle with shape A000041 defined by sequence A098546 times sequence A036040.
Original entry on oeis.org
1, 2, 1, 3, 9, 1, 4, 24, 18, 24, 1, 5, 50, 100, 100, 150, 50, 1, 6, 90, 225, 150, 300, 1200, 300, 300, 675, 90, 1, 7, 147, 441, 735, 735, 3675, 2450, 3675, 1225, 7350, 3675, 735, 2205, 147, 1, 8, 224, 784, 1568, 980, 1568, 9408, 15680, 11760, 15680, 3920, 29400
Offset: 1
A098546(n) begins 1 2 1 3 3 1 4 6 6 4 1 ...
A036040(n) begins 1 1 1 1 3 1 1 4 3 6 1 ...
So the triangle begins:
1;
2, 1;
3, 9, 1;
4, 24, 18, 24, 1;
5, 50, 100, 100, 150, 50, 1;
6, 90, 225, 150, 300, 1200, 300, 300, 675, 90, 1;
7, 147, 441, 735, 735, 3675, 2450, 3675, 1225, 7350, 3675, 735, 2205, 147, 1;
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sortAbrSteg := proc(L1,L2) local i ; if nops(L1) < nops(L2) then RETURN(true) ; elif nops(L2) < nops(L1) then RETURN(false) ; else for i from 1 to nops(L1) do if op(i,L1) < op(i,L2) then RETURN(false) ; fi ; od ; RETURN(true) ; fi ; end: A098546 := proc(n,k) local prts,m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then m := nops(op(k,prts)) ; binomial(n,m) ; else 0 ; fi ; end: M3 := proc(L) local n,k,an,resul; n := add(i,i=L) ; resul := factorial(n) ; for k from 1 to n do an := add(1-min(abs(j-k),1),j=L) ; resul := resul/ (factorial(k))^an /factorial(an) ; od ; end: A036040 := proc(n,k) local prts,m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then M3(op(k,prts)) ; else 0 ; fi ; end: A122454 := proc(n,k) A098546(n,k)*A036040(n,k) ; end: for n from 1 to 10 do for k from 1 to combinat[numbpart](n) do a:=A122454(n,k) ; printf("%d, ",a) ; od; od ; # R. J. Mathar, Jul 17 2007
A343841
a(n) = Sum{k=0..n} (-1)^(n-k)*binomial(n, k)*Stirling2(n, k).
Original entry on oeis.org
1, 1, -1, -5, 15, 56, -455, -237, 16947, -64220, -529494, 6833608, -8606015, -459331677, 4335744673, 6800310151, -518075832085, 4315086396640, 19931595013738, -812870258798156, 6648395876520816, 46852711038750520, -1752440325584024944, 15485712825845269456
Offset: 0
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a := n -> add((-1)^(n-k)*binomial(n, k)*Stirling2(n, k), k=0..n):
seq(a(n), n = 0..24);
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a[n_] := Sum[(-1)^(n - k) * Binomial[n, k] * StirlingS2[n, k], {k, 0, n}]; Array[a, 24, 0] (* Amiram Eldar, May 07 2021 *)
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a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*stirling(n, k, 2)); \\ Michel Marcus, May 07 2021
A271702
Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j-1,-n-1)*S2(k,j), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 1, 3, 6, 13, 1, 4, 10, 26, 71, 1, 5, 15, 45, 140, 456, 1, 6, 21, 71, 246, 887, 3337, 1, 7, 28, 105, 399, 1568, 6405, 27203, 1, 8, 36, 148, 610, 2584, 11334, 51564, 243203, 1, 9, 45, 201, 891, 4035, 18849, 91101, 455712, 2357356
Offset: 0
Triangle starts:
[1]
[1, 1]
[1, 2, 3]
[1, 3, 6, 13]
[1, 4, 10, 26, 71]
[1, 5, 15, 45, 140, 456]
[1, 6, 21, 71, 246, 887, 3337]
[1, 7, 28, 105, 399, 1568, 6405, 27203]
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T := (n,k) -> add(Stirling2(k,j)*binomial(-j-1,-n-1)*(-1)^(n-j),j=0..n):
seq(seq(T(n,k), k=0..n), n=0..9);
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Flatten[Table[Sum[(-1)^(n-j) Binomial[-j-1,-n-1] StirlingS2[k,j], {j,0,n}], {n,0,9}, {k,0,n}]]
A344053
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*Stirling2(n, k)*k!.
Original entry on oeis.org
1, 1, 0, -9, -40, 125, 3444, 18571, -241872, -5796711, -24387220, 1132278191, 25132445832, 8850583573, -10681029498972, -214099676807085, 1643397436986464, 176719161389104817, 2976468247699317468, -71662294521163070153, -4638920054290748840520, -55645074852328083377619
Offset: 0
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a[n_] := Sum[(-1)^(n - k) * Binomial[n, k] * StirlingS2[n, k] * k!, {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, May 10 2021 *)
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a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*stirling(n, k, 2)*k!); \\ Michel Marcus, May 10 2021
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