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A372921
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Triangle read by rows: T(n, k) = (Sum_{i=0..n-k} (-1)^i * binomial(n-k, i) * A007559(n-i)) * n! / ((n-k)! * A007559(k)) for 0 <= k <= n.
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0
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1, 0, 1, 3, 6, 2, 18, 63, 36, 6, 189, 828, 684, 216, 24, 2484, 13365, 14400, 6660, 1440, 120, 40095, 255474, 339390, 206280, 65880, 10800, 720, 766422, 5645619, 8915508, 6707610, 2827440, 687960, 90720, 5040, 16936857, 141626232, 259137144, 232306704, 121519440, 39130560, 7680960, 846720, 40320
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OFFSET
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0,4
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LINKS
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FORMULA
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T(n, k) = (T(n-1, k-1) + 3 * T(n-1, k)) * n for 0 < k < n with initial values T(n, 0) = A033030(n) and T(n, n) = A000142(n).
E.g.f. of column k: exp(-t) / (1-3*t)^(1/3) * (t / (1-3*t))^k.
E.g.f.: exp(x*t / (1-3*t) - t) / (1-3*t)^(1/3).
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EXAMPLE
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Triangle T(n, k) starts:
n\k : 0 1 2 3 4 5 6 7
======================================================================
0 : 1
1 : 0 1
2 : 3 6 2
3 : 18 63 36 6
4 : 189 828 684 216 24
5 : 2484 13365 14400 6660 1440 120
6 : 40095 255474 339390 206280 65880 10800 720
7 : 766422 5645619 8915508 6707610 2827440 687960 90720 5040
etc.
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MATHEMATICA
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T[n_, k_]:=n!SeriesCoefficient[Exp[-t]/ (1-3*t)^(1/3) * (t / (1-3*t))^k, {t, 0, n}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}]//Flatten (* Stefano Spezia, May 18 2024 *)
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PROG
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(PARI) T(n, k) = { sum(i=0, n-k, (-1)^i * binomial(n-k, i) * prod(j=1, n-i, 3*j-2)) * n! / ((n-k)! * prod(m=1, k, 3*m-2)) }
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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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