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A212207
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Triangle read by rows: coefficients of polynomials p_{n,n-1}(x) arising in enumeration of two-line arrays.
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0
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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
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OFFSET
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0,5
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COMMENTS
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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.
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LINKS
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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).
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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Cf. A001263.
Sequence in context: A227790 A181897 A337977 * A111049 A211955 A088617
Adjacent sequences: A212204 A212205 A212206 * A212208 A212209 A212210
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KEYWORD
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nonn,tabl
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AUTHOR
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N. J. A. Sloane, May 15 2012
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STATUS
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approved
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