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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 Narayama numbers A001263.
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LINKS
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Table of n, a(n) for n=0..58.
Carlitz, L. and Riordan, John. 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)/(1-N(x,y)^2), where N(x,y) is g.f. of Narayana numbers (A001263). -Vladimir Kruchinin, Apr 10 2018
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EXAMPLE
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Triangle begins:
1
1 1
1 3 2
1 6 9 4
1 10 26 25 8
...
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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 */
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CROSSREFS
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Cf. A001263.
Sequence in context: A052174 A227790 A181897 * 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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