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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.
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%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