A212207 Triangle read by rows: coefficients of polynomials p_{n,n-1}(x) arising in enumeration of two-line arrays.

1, 1, 1, 1, 3, 2, 1, 6, 9, 4, 1, 10, 26, 25, 8, 1, 15, 60, 95, 65, 16, 1, 21, 120, 280, 309, 161, 32, 1, 28, 217, 700, 1113, 924, 385, 64, 1, 36, 364, 1554, 3346, 3948, 2596, 897, 128, 1, 45, 576, 3150, 8820, 13902, 12864, 6957, 2049, 256, 1, 55, 870, 5940

Offset

0,5

Comments

These polynomials are defined in Section 3 of Carlitz-Riordan (1971). Equation (3.14) claims to be a recurrence, which unfortunately I could not get to work. The coefficients of the polynomials A_n(x) = a_{n,n}(x) which appear in (3.14) are the Narayana numbers A001263.

Links

Table of n, a(n) for n=0..58.

L. Carlitz and J. Riordan, Enumeration of some two-line arrays by extent. J. Combinatorial Theory Ser. A 10 1971 271--283. MR0274301(43 #66).

L. Carlitz and J. Riordan, Enumeration of some two-line arrays by extent, J. Combinatorial Theory Ser. A 10 1971 271-283 (MR274301 Review by Richard P. Stanley).

Formula

G.f.: N(x,y)/(x*y*(1-N(x,y)^2)), where N(x,y) is g.f. of Narayana numbers (A001263). - Vladimir Kruchinin, Apr 10 2018, corrected by Yuriy Shablya, May 05 2021

T(n,m) = Sum_{k=0..m} ((k+1)/(n+1))*binomial(n+1,m+1)*binomial(n+1,m-k)*((1+(-1)^k)/2). - Yuriy Shablya, May 05 2021

Example

Triangle begins:
---------------------------------------------------------------------
n \ m |     0     1     2     3     4     5     6     7     8     9
-------+-------------------------------------------------------------
   0   |     1
   1   |     1     1
   2   |     1     3     2
   3   |     1     6     9     4
   4   |     1    10    26    25     8
   5   |     1    15    60    95    65    16
   6   |     1    21   120   280   309   161    32
   7   |     1    28   217   700  1113   924   385    64
   8   |     1    36   364  1554  3346  3948  2596   897   128
   9   |     1    45   576  3150  8820 13902 12864  6957  2049   256

Mathematica

Table[Sum[((k + 1)/(n + 1))*Binomial[n + 1, m + 1] Binomial[n + 1, m - k]*((1 + (-1)^k)/2), {k, 0, m}], {n, 0, 10}, {m, 0, n}] // Flatten (* Michael De Vlieger, May 07 2021 *)

Prog

(PARI)
{T(n, k) = if( n < k || k < 0, 0, sum( j=0, k, binomial( n+1, k+1) * binomial( n+1, k-j) * if( j%2, -(n+1 +j-k), k+1)) / (n+1))} /* Michael Somos, Aug 22 2012 */
(Maxima)
T(n, m):=sum(((k+1)/(n+1))*binomial(n+1, m+1)*binomial(n+1, m-k)*((1+(-1)^k)/2), k, 0, m) /* Yuriy Shablya, May 05 2021 */

Crossrefs

Cf. A001263.

Sequence in context: A227790 A181897 A337977 * A111049 A211955 A088617

Adjacent sequences: A212204 A212205 A212206 * A212208 A212209 A212210

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 15 2012

Status

approved