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%I #41 Jun 06 2021 17:36:48
%S 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,
%T 161,32,1,28,217,700,1113,924,385,64,1,36,364,1554,3346,3948,2596,897,
%U 128,1,45,576,3150,8820,13902,12864,6957,2049,256,1,55,870,5940
%N Triangle read by rows: coefficients of polynomials p_{n,n-1}(x) arising in enumeration of two-line arrays.
%C 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.
%H L. Carlitz and J. Riordan, <a href="http://dx.doi.org/10.1016/0097-3165(71)90032-X">Enumeration of some two-line arrays by extent</a>. J. Combinatorial Theory Ser. A 10 1971 271--283. MR0274301(43 #66).
%H L. Carlitz and J. Riordan, <a href="http://www.ams.org/mathscinet-getitem?mr=274301">Enumeration of some two-line arrays by extent</a>, J. Combinatorial Theory Ser. A 10 1971 271-283 (MR274301 Review by Richard P. Stanley).
%F 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
%F 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
%e Triangle begins:
%e ---------------------------------------------------------------------
%e n \ m | 0 1 2 3 4 5 6 7 8 9
%e -------+-------------------------------------------------------------
%e 0 | 1
%e 1 | 1 1
%e 2 | 1 3 2
%e 3 | 1 6 9 4
%e 4 | 1 10 26 25 8
%e 5 | 1 15 60 95 65 16
%e 6 | 1 21 120 280 309 161 32
%e 7 | 1 28 217 700 1113 924 385 64
%e 8 | 1 36 364 1554 3346 3948 2596 897 128
%e 9 | 1 45 576 3150 8820 13902 12864 6957 2049 256
%t 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 *)
%o (PARI)
%o {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 */
%o (Maxima)
%o 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 */
%Y Cf. A001263.
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_, May 15 2012
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