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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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%I #11 Jan 05 2024 14:56:56

%S 1,1,2,2,10,10,6,52,120,80,24,280,1160,1760,880,120,1520,10000,27200,

%T 30800,12320,720,11280,78160,343200,695200,628320,209440,5040,164640,

%U 784000,3684800,12073600,19490240,14660800,4188800,40320,1438080,15532160,48294400,170755200,445688320,598160640,385369600,96342400

%N 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.

%H G. C. Greubel, <a href="/A168295/b168295.txt">Rows n = 1..50 of the triangle, flattened</a>

%F 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).

%F From _G. C. Greubel_, Mar 17 2022: (Start)

%F 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).

%F T(n, 1) = (n-1)!.

%F T(n, n) = A008544(n-1). (End)

%e Triangle begins as:

%e 1;

%e 1, 2;

%e 2, 10, 10;

%e 6, 52, 120, 80;

%e 24, 280, 1160, 1760, 880;

%e 120, 1520, 10000, 27200, 30800, 12320;

%e 720, 11280, 78160, 343200, 695200, 628320, 209440;

%e 5040, 164640, 784000, 3684800, 12073600, 19490240, 14660800, 4188800;

%t 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]];

%t A142458[n_, k_]:= A142458[n, k]= T[n,k,3];

%t p[x_, n_]:= p[x, n]= Sum[A142458[n,k]*Pochhammer[x+k-n+1, n-1], {k, n}];

%t Table[CoefficientList[p[x, n], x], {n, 1, 12}]//Flatten (* modified by _G. C. Greubel_, Mar 17 2022 *)

%o (Sage)

%o @CachedFunction

%o def T(n,k,m): # A142458

%o if (k==1 or k==n): return 1

%o else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)

%o def A142458(n,k): return T(n,k,3)

%o @CachedFunction

%o def p(n,x): return sum( A142458(n,j)*rising_factorial(x+j-n+1, n-1) for j in (1..n))

%o def A168295(n,k): return ( p(n,x) ).series(x,n+1).list()[k-1]

%o flatten([[ A168295(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 17 2022

%Y Cf. A008544, A142458, A167786.

%K sign,tabl

%O 1,3

%A _Roger L. Bagula_, Nov 22 2009

%E Edited by _G. C. Greubel_, Mar 17 2022