A168391 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.

1, 1, 2, 2, 5, 5, 6, 19, 21, 14, 24, 84, 126, 84, 42, 120, 468, 750, 720, 330, 132, 720, 2988, 5496, 5445, 3795, 1287, 429, 5040, 22356, 43120, 50435, 35035, 19019, 5005, 1430, 40320, 186912, 391688, 472472, 398398, 208208, 92092, 19448, 4862

Offset

1,3

Comments

The Z-transform of the triangle gives the Narayan triangle (A001263).

Every other polynomial, of p(n, x), has a factor of (1+2*x), just like the higher Sierpinski-Pascal Worpitzky form polynomials.

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

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).

From G. C. Greubel, Mar 28 2022: (Start)

Sum_{k=0..n-1} T(n, k) = A001710(n+1).

T(n, 0) = n!. (End)

Example

Triangle begins as:
      1;
      1,      2;
      2,      5,      5;
      6,     19,     21,     14;
     24,     84,    126,     84,     42;
    120,    468,    750,    720,    330,    132;
    720,   2988,   5496,   5445,   3795,   1287,   429;
   5040,  22356,  43120,  50435,  35035,  19019,  5005,  1430;
  40320, 186912, 391688, 472472, 398398, 208208, 92092, 19448, 4862;

Mathematica

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}];
A168391[n_]:= CoefficientList[p[x, n], x];
Table[A168391[n], {n, 12}]//Flatten (* G. C. Greubel, Mar 28 2022 *)

Prog

(Sage)
@CachedFunction
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) )
def A168391(n, k): return ( p(n, x) ).series(x, n+1).list()[k]
flatten([[A168391(n, k) for k in (0..n-1)] for n in (1..12)]) # G. C. Greubel, Mar 28 2022

Crossrefs

Cf. A001263, A001710.

Sequence in context: A069896 A053246 A219651 * A157123 A265764 A308768

Adjacent sequences: A168388 A168389 A168390 * A168392 A168393 A168394

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 24 2009

Extensions

Edited by G. C. Greubel, Mar 28 2022

Status

approved