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

0,1

Links

Table of n, a(n) for n=0..54.

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

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

Example

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;
    ...

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 *)

Prog

(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 */

Crossrefs

Row sums: 2*A000108 (without the first term).

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

Sequence in context: A098745 A029158 A241953 * A155162 A275444 A243322

Adjacent sequences: A156194 A156195 A156196 * A156198 A156199 A156200

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Feb 05 2009

Extensions

Edited by Franck Maminirina Ramaharo, Dec 11 2018

Status

approved