A156197 T(n,k) = A009766(n,k) + A009766(n,n-k), triangle read by rows.

2, 2, 2, 3, 4, 3, 6, 8, 8, 6, 15, 18, 18, 18, 15, 43, 47, 42, 42, 47, 43, 133, 138, 110, 96, 110, 138, 133, 430, 436, 324, 240, 240, 324, 436, 430, 1431, 1438, 1036, 682, 550, 682, 1036, 1438, 1431, 4863, 4871, 3476, 2156, 1430, 1430, 2156, 3476, 4871, 4863

Offset: 0

Author

Roger L. Bagula, Feb 05 2009

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins:
      2;
      2,    2;
      3,    4,    3;
      6,    8,    8,    6;
     15,   18,   18,   18,   15;
     43,   47,   42,   42,   47,   43;
    133,  138,  110,   96,  110,  138,  133;
    430,  436,  324,  240,  240,  324,  436,  430;
   1431, 1438, 1036,  682,  550,  682, 1036, 1438, 1431;
   4863, 4871, 3476, 2156, 1430, 1430, 2156, 3476, 4871, 4863;
    ...
		

Links

• L. Carlitz and J. Riordan, Two element lattice permutation numbers and their q-generalization, Duke Math. J. Volume 31, Number 3 (1964), 371-388.

Crossrefs

Row sums: 2*A000108(n+1).

Cf. A009766, A033184. - Roger L. Bagula and Gary W. Adamson, Dec 03 2009.

Programs

Mathematica

t0[n_, m_] = Binomial[n + m, n] - Binomial[n + m, m - 1];
T[n_, m_] = FullSimplify[t0[n, m] + t0[n, n - m]];
Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}] // Flatten
(* or *)
Table[Table[((1 - k + n)*Binomial[k + n, n] + (1 + k)*Binomial[-k + 2*n, n])/(1 + n), {k, 0, n}], {n, 0, 10}] // Flatten (* Roger L. Bagula and Gary W. Adamson, Dec 03 2009 *)

Maxima

A009766(n, k) := binomial(n + k, n)*(n - k + 1)/(n + 1)$
create_list(A009766(n, k) + A009766(n, n - k), n, 0, 10, k, 0, n); /* Franck Maminirina Ramaharo, Dec 11 2018 */

Formula

T(n,k) = -binomial(k + n, -1 + k) + binomial(k + n, n) + binomial(-k + 2*n, n) - binomial(-k + 2*n, -1 - k + n).

From Roger L. Bagula and Gary W. Adamson, Dec 03 2009: (Start)

T(n,k) = ((n - k + 1)*binomial(n + k, n) + (k + 1)*binomial(-k + 2*n, n))/(n + 1).

T(n,k) = A009766(n,k) + A033184(n,k). (End)

G.f.: (C(t*x) + C(x)*(1 - x*C(t*x) - t*x*C(t*x)))/((1 - t*x*C(x))*(1 - x*C(t*x))), where C(x) = (1 - sqrt(1 - 4*x))/(2*x). - Franck Maminirina Ramaharo, Dec 11 2018

Extensions

Edited by Franck Maminirina Ramaharo, Dec 11 2018