%I #7 Mar 29 2022 01:26:01
%S 1,1,2,2,5,5,6,19,21,14,24,84,126,84,42,120,468,750,720,330,132,720,
%T 2988,5496,5445,3795,1287,429,5040,22356,43120,50435,35035,19019,5005,
%U 1430,40320,186912,391688,472472,398398,208208,92092,19448,4862
%N Triangle T(n, k) = [x^k]( p(n,x) ), where p(n, x) = Sum_{k=1..n} A001263(n,k)*binomial(x+k -1, n-1), read by rows.
%C The Z-transform of the triangle gives the Narayan triangle (A001263).
%C Every other polynomial, of p(n, x), has a factor of (1+2*x), just like the higher Sierpinski-Pascal Worpitzky form polynomials.
%H G. C. Greubel, <a href="/A168391/b168391.txt">Rows n = 1..50 of the triangle, flattened</a>
%F T(n, k) = [x^k]( p(n,x) ), where p(n, x) = Sum_{k=1..n} A001263(n,k)*binomial(x+k -1, n-1).
%F From _G. C. Greubel_, Mar 28 2022: (Start)
%F Sum_{k=0..n-1} T(n, k) = A001710(n+1).
%F T(n, 0) = n!. (End)
%e Triangle begins as:
%e 1;
%e 1, 2;
%e 2, 5, 5;
%e 6, 19, 21, 14;
%e 24, 84, 126, 84, 42;
%e 120, 468, 750, 720, 330, 132;
%e 720, 2988, 5496, 5445, 3795, 1287, 429;
%e 5040, 22356, 43120, 50435, 35035, 19019, 5005, 1430;
%e 40320, 186912, 391688, 472472, 398398, 208208, 92092, 19448, 4862;
%t p[n_, x_]:= p[n,x]= ((-1)^(n+1)/(n+1))*Sum[Binomial[n-1, k-1]*Binomial[n+1, k]*Pochhammer[1-k-x, n-1], {k, n}];
%t A168391[n_]:= CoefficientList[p[x, n], x];
%t Table[A168391[n], {n,12}]//Flatten (* _G. C. Greubel_, Mar 28 2022 *)
%o (Sage)
%o @CachedFunction
%o def p(n,x): return ((-1)^(n+1)/(n+1))*sum( binomial(n+1, k)*binomial(n-1, k-1)*rising_factorial(1-k-x, n-1) for k in (1..n) )
%o def A168391(n,k): return ( p(n,x) ).series(x, n+1).list()[k]
%o flatten([[A168391(n,k) for k in (0..n-1)] for n in (1..12)]) # _G. C. Greubel_, Mar 28 2022
%Y Cf. A001263, A001710.
%K nonn,tabl
%O 1,3
%A _Roger L. Bagula_, Nov 24 2009
%E Edited by _G. C. Greubel_, Mar 28 2022