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A331333
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Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= k <= n.
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3
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1, 1, 2, 2, 8, 4, 6, 36, 36, 8, 24, 192, 288, 128, 16, 120, 1200, 2400, 1600, 400, 32, 720, 8640, 21600, 19200, 7200, 1152, 64, 5040, 70560, 211680, 235200, 117600, 28224, 3136, 128, 40320, 645120, 2257920, 3010560, 1881600, 602112, 100352, 8192, 256
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OFFSET
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0,3
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LINKS
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FORMULA
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T(n, k) = n!*S(n, k) where S(n, k) is recursively defined by:
if k = 0 then 1 else if k > n then 0 else 2*S(n-1, k-1)/k + S(n-1, k).
T(n,k) = 2^k*(n!/k!)*binomial(n,k).
E.g.f.: 1/ (1 - x)*exp(2*x*t)/(1 - x)) = 1 + (1 + 2*t)*x + (2 + 8*t + 4*t^2)*x^2/2! + .... Cf. A021009. (End)
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EXAMPLE
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Triangle starts:
[0] 1
[1] 1, 2
[2] 2, 8, 4
[3] 6, 36, 36, 8
[4] 24, 192, 288, 128, 16
[5] 120, 1200, 2400, 1600, 400, 32
[6] 720, 8640, 21600, 19200, 7200, 1152, 64
[7] 5040, 70560, 211680, 235200, 117600, 28224, 3136, 128
[8] 40320, 645120, 2257920, 3010560, 1881600, 602112, 100352, 8192, 256
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MAPLE
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A331333 := proc(n, k) local S; S := proc(n, k) option remember;
`if`(k = 0, 1, `if`(k > n, 0, 2*S(n-1, k-1)/k + S(n-1, k))) end: n!*S(n, k) end:
seq(seq(A331333(n, k), k=0..n), n=0..8);
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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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