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A131689
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Triangle of numbers T(n,k)=k!*Stirling2(n,k)=A000142(k)*A048993(n,k) read by rows (n>=0,0<=k<=n).
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10
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1, 0, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 14, 36, 24, 0, 1, 30, 150, 240, 120, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 126, 1806, 8400, 16800, 15120, 5040, 0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320, 0, 1, 510, 18150, 186480, 834120, 1905120, 2328480
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,0,2,0,3,0,4,0,5,0,6,0,7,0,...] DELTA [1,1,2,2,3,3,4,4,5,5,6,6,...] where DELTA is the operator defined in A084938 ; another version of A019538 .
See also A019538 : version with n>0 and k>0. [From Philippe DELEHAM, Nov 03 2008]
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FORMULA
| T(n,k)=k!*(T(n-1,k-1)+T(n-1,k)) with T(0,0)=1. Sum_{k, 0<=k<=n}T(n,k)*x^k = (-1)^n*A000629(n), A033999(n), A000007(n), A000670(n), A004123(n+1), A032033(n), A094417(n), A094418(n), A094419(n) for x=-2, -1, 0, 1, 2, 3, 4, 5, 6 respectively .
Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)=A000012(n), A000142(n), A000670(n), A122704(n) for x=-1, 0, 1, 2 respectively . - Philippe DELEHAM, Oct 09 2007
Sum_{0<=k<=n} (-1)^k*T(n,k)/(k+1) = Bernoulli numbers A027641(n)/A027642(n). - Peter Luschny, Sep 17 2011
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EXAMPLE
| Triangle begins:
1;
0, 1;
0, 1, 2;
0, 1, 6, 6;
0, 1, 14, 36, 24;
0, 1, 30, 150, 240, 120;
0, 1, 62, 540, 1560, 1800, 720;
0, 1, 126, 1806, 8400, 16800, 15120, 5040;
0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320;
0, 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880 ;...
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MAPLE
| A131689 := proc(n, k) combinat[stirling2](n, k)*k! end: # Peter Luschny, Sep 17 2011
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CROSSREFS
| Cf. Diagonals : A000142, A001286, A037960, A037961, A037962, A037963.
Sequence in context: A089949 A085845 A138106 * A114329 A101371 A154974
Adjacent sequences: A131686 A131687 A131688 * A131690 A131691 A131692
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KEYWORD
| nonn,tabl
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 14 2007
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