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A168295
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Triangle T(n, k) = coefficients of (p(x,n)), where p(x, n) = (n-1)! * Sum_{j=1..n} A142458(n, j)*binomial(x+j-1, n-1), read by rows.
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1
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1, 1, 2, 2, 10, 10, 6, 52, 120, 80, 24, 280, 1160, 1760, 880, 120, 1520, 10000, 27200, 30800, 12320, 720, 11280, 78160, 343200, 695200, 628320, 209440, 5040, 164640, 784000, 3684800, 12073600, 19490240, 14660800, 4188800, 40320, 1438080, 15532160, 48294400, 170755200, 445688320, 598160640, 385369600, 96342400
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OFFSET
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1,3
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LINKS
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FORMULA
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T(n, k) = coefficients of (p(x,n)), where p(x, n) = (n-1)! * Sum_{j=1..n} A142458(n, j)*binomial(x+j-1, n-1).
T(n, k) = coefficients of (p(n, x)), where p(n, x) = Sum_{j=1..n} A142458(n, j)*Pochhammer(x+j-n+1, n-1).
T(n, 1) = (n-1)!.
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EXAMPLE
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Triangle begins as:
1;
1, 2;
2, 10, 10;
6, 52, 120, 80;
24, 280, 1160, 1760, 880;
120, 1520, 10000, 27200, 30800, 12320;
720, 11280, 78160, 343200, 695200, 628320, 209440;
5040, 164640, 784000, 3684800, 12073600, 19490240, 14660800, 4188800;
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MATHEMATICA
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T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k-m+1)*T[n-1, k, m]];
p[x_, n_]:= p[x, n]= Sum[A142458[n, k]*Pochhammer[x+k-n+1, n-1], {k, n}];
Table[CoefficientList[p[x, n], x], {n, 1, 12}]//Flatten (* modified by G. C. Greubel, Mar 17 2022 *)
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PROG
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(Sage)
@CachedFunction
if (k==1 or k==n): return 1
else: return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m)
def A142458(n, k): return T(n, k, 3)
@CachedFunction
def p(n, x): return sum( A142458(n, j)*rising_factorial(x+j-n+1, n-1) for j in (1..n))
def A168295(n, k): return ( p(n, x) ).series(x, n+1).list()[k-1]
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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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