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A156584
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Triangle T(n,k) = SF(n+1)/(SF(n-k+1)*SF(k+1)) where SF(n) is the superfactorial A000178(n), read by rows.
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2
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1, 1, 1, 1, 3, 1, 1, 12, 12, 1, 1, 60, 240, 60, 1, 1, 360, 7200, 7200, 360, 1, 1, 2520, 302400, 1512000, 302400, 2520, 1, 1, 20160, 16934400, 508032000, 508032000, 16934400, 20160, 1, 1, 181440, 1219276800, 256048128000, 1536288768000, 256048128000, 1219276800, 181440, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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T(n, k) = BarnesG(n+3)/(BarnesG(k+3)*BarnesG(n-k+3)).
T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), with T(0, k, m) = 1, f(n, k) = (-1)^n*(n + 1)!*BarnesG(n+k+1)/(Gamma(k+1)^n*BarnesG(k+1)), f(n, 0) = n!, and m = 1. (End)
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EXAMPLE
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Triangle begins as:
1;
1, 1;
1, 3, 1;
1, 12, 12, 1;
1, 60, 240, 60, 1;
1, 360, 7200, 7200, 360, 1;
1, 2520, 302400, 1512000, 302400, 2520, 1;
1, 20160, 16934400, 508032000, 508032000, 16934400, 20160, 1;
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MAPLE
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SF := n -> mul(j!, j=0..n): T := (n, k) -> SF(n-1)/(SF(n-k)*SF(k)):
seq(print(seq(T(n, k), k=1..n-1)), n=0..9); # Peter Luschny, Jan 24 2015
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MATHEMATICA
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(* First program *)
b[n_, k_]:= If[k==0, n!, Product[Sum[(-1)^(i+j)*(j+1)*StirlingS1[j-1, i]*(k+1)^i, {i, 0, j-1}], {j, 1, n}]];
T[n_, k_, m_] = If[n==0, 1, b[n, m]/(b[k, m]*b[n-k, m])];
Table[T[n, k, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 20 2021 *)
(* Second program *)
f[n_, k_]:= If[k==0, n!, (-1)^n*(n+1)!*BarnesG[n+k+1]/(Gamma[k+1]^n*BarnesG[k+1])];
T[n_, k_, m_]:= If[n==0, 1, f[n, m]/(f[k, m]*f[n-k, m])];
Table[T[n, k, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 20 2021 *)
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PROG
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(Sage)
def f(n, k): return factorial(n) if (k==0) else (-1)^n*factorial(n+1)*product( rising_factorial(k+1, j) for j in (0..n-1) )
def T(n, k, m): return 1 if (n==0) else f(n, m)/(f(k, m)*f(n-k, m))
flatten([[T(n, k, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 21 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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