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

0,1

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.

Links

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

Formula

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

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

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A143491.

Sequence in context: A045683 A343031 A157531 * A080088 A098715 A167886

Adjacent sequences: A155752 A155753 A155754 * A155756 A155757 A155758

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 26 2009

Status

approved