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A168295 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.

%I A168295 #11 Jan 05 2024 14:56:56
%S A168295 1,1,2,2,10,10,6,52,120,80,24,280,1160,1760,880,120,1520,10000,27200,
%T A168295 30800,12320,720,11280,78160,343200,695200,628320,209440,5040,164640,
%U A168295 784000,3684800,12073600,19490240,14660800,4188800,40320,1438080,15532160,48294400,170755200,445688320,598160640,385369600,96342400
%N A168295 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 A168295 G. C. Greubel, <a href="/A168295/b168295.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A168295 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 A168295 From _G. C. Greubel_, Mar 17 2022: (Start)
%F A168295 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 A168295 T(n, 1) = (n-1)!.
%F A168295 T(n, n) = A008544(n-1). (End)
%e A168295 Triangle begins as:
%e A168295      1;
%e A168295      1,      2;
%e A168295      2,     10,     10;
%e A168295      6,     52,    120,      80;
%e A168295     24,    280,   1160,    1760,      880;
%e A168295    120,   1520,  10000,   27200,    30800,    12320;
%e A168295    720,  11280,  78160,  343200,   695200,   628320,   209440;
%e A168295   5040, 164640, 784000, 3684800, 12073600, 19490240, 14660800, 4188800;
%t A168295 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 A168295 A142458[n_, k_]:= A142458[n, k]= T[n,k,3];
%t A168295 p[x_, n_]:= p[x, n]= Sum[A142458[n,k]*Pochhammer[x+k-n+1, n-1], {k, n}];
%t A168295 Table[CoefficientList[p[x, n], x], {n, 1, 12}]//Flatten (* modified by _G. C. Greubel_, Mar 17 2022 *)
%o A168295 (Sage)
%o A168295 @CachedFunction
%o A168295 def T(n,k,m): # A142458
%o A168295     if (k==1 or k==n): return 1
%o A168295     else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
%o A168295 def A142458(n,k): return T(n,k,3)
%o A168295 @CachedFunction
%o A168295 def p(n,x): return sum( A142458(n,j)*rising_factorial(x+j-n+1, n-1) for j in (1..n))
%o A168295 def A168295(n,k): return ( p(n,x) ).series(x,n+1).list()[k-1]
%o A168295 flatten([[ A168295(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 17 2022
%Y A168295 Cf. A008544, A142458, A167786.
%K A168295 sign,tabl
%O A168295 1,3
%A A168295 _Roger L. Bagula_, Nov 22 2009
%E A168295 Edited by _G. C. Greubel_, Mar 17 2022