A155755 Triangle T(n, k) = A143491(n+2, k+2) + A143491(n+2, n-k+2), read by rows.

2, 3, 3, 7, 10, 7, 25, 35, 35, 25, 121, 168, 142, 168, 121, 721, 1064, 735, 735, 1064, 721, 5041, 8055, 5399, 3330, 5399, 8055, 5041, 40321, 69299, 49371, 22449, 22449, 49371, 69299, 40321, 362881, 663740, 509830, 223300, 109298, 223300, 509830, 663740, 362881

Offset: 0

Author

Roger L. Bagula, Jan 26 2009

Keywords

• nonn
• tabl

Comments

This symmetric summation of the triangle A143491 is equivalent to the coefficient [x^m] (p_n(x) + x^n*p_n(1/x)) of the polynomials defined in A143491 plus their reverses.

Examples

			Triangle begins as:
       2;
       3,      3;,
       7,     10,      7;
      25,     35,     35,     25;
     121,    168,    142,    168,    121;
     721,   1064,    735,    735,   1064,    721;
    5041,   8055,   5399,   3330,   5399,   8055,   5041;
   40321,  69299,  49371,  22449,  22449,  49371,  69299,  40321;
  362881, 663740, 509830, 223300, 109298, 223300, 509830, 663740, 362881;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A143491.

Programs

Mathematica

(* First program *)
q[x_, n_]:= Product[x +n-i+1, {i,0,n-1}];
p[x_, n_]:= q[x, n] + x^n*q[1/x, n];
Table[CoefficientList[p[x, n], x], {n,0,12}]//Flatten (* modified by G. C. Greubel, Jun 06 2021 *)
(* Second program *)
A143491[n_, k_]:= (n-2)!*Sum[(n-k-j+1)*Abs[StirlingS1[j+k-2, k-2]]/(j+k-2)!, {j,0,n-k}];
A155755[n_, k_]:= A143491[n+2, k+2] + A143491[n+2, n-k+2];
Table[A155755[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 06 2021 *)

Sage

def A143491(n,k): return factorial(n-2)*sum( (n-k-j+1)*stirling_number1(j+k-2, k-2)/factorial(j+k-2) for j in (0..n-k) )
def A155755(n,k): return A143491(n+2, k+2) + A143491(n+2, n-k+2)
flatten([[A155755(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 06 2021

Formula

T(n, k) = A143491(n+2, k+2) + A143491(n+2, n-k+2).

Sum_{k=0..n} T(n, k) = (n+2)!.