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A269948
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Triangle read by rows, Stirling set numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+k^3*T(n-1, k), for n>=0 and 0<=k<=n.
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3
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1, 0, 1, 0, 1, 1, 0, 1, 9, 1, 0, 1, 73, 36, 1, 0, 1, 585, 1045, 100, 1, 0, 1, 4681, 28800, 7445, 225, 1, 0, 1, 37449, 782281, 505280, 35570, 441, 1, 0, 1, 299593, 21159036, 33120201, 4951530, 130826, 784, 1
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OFFSET
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0,9
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COMMENTS
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Also called 3rd central factorial numbers.
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LINKS
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FORMULA
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T(n,2) = (8^(n-1)-1)/7 for n>=1 (cf. A023001).
T(n,n-1) = (n*(n-1)/2)^2 for n>=1 (cf. A000537).
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EXAMPLE
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1,
0, 1,
0, 1, 1,
0, 1, 9, 1,
0, 1, 73, 36, 1,
0, 1, 585, 1045, 100, 1,
0, 1, 4681, 28800, 7445, 225, 1,
0, 1, 37449, 782281, 505280, 35570, 441, 1.
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MAPLE
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T := proc(n, k) option remember;
`if`(n=k, 1,
`if`(k<0 or k>n, 0,
T(n-1, k-1) + k^3*T(n-1, k))) end:
for n from 0 to 9 do seq(T(n, k), k=0..n) od;
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MATHEMATICA
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T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + k^3*T[n - 1, k]; T[_, _] = 0;
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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