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A212207 Triangle read by rows: coefficients of polynomials p_{n,n-1}(x) arising in enumeration of two-line arrays. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified October 5 13:40 EDT 2022. Contains 357258 sequences. (Running on oeis4.)