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A156715 Symmetrical version of Graham and Riordan 1966:k=1; t(n,m,k)=(((2*k + 1)/(m + k + 1))*Binomial[n - 1 - k, m - k]*Binomial[n + k, m + k] + ((2*k + 1)/( n - m - 1 + k + 1))*Binomial[n - 1 - k, n - m - 1 - k]*Binomial[ n + k, n - m - 1 + k]) 0

%I

%S 3,3,3,12,3,3,25,25,3,3,42,90,42,3,3,63,231,231,63,3,3,88,490,840,490,

%T 88,3,3,117,918,2394,2394,918,117,3,3,150,1575,5796,8820,5796,1575,

%U 150,3,3,187,2530,12474,26796,26796,12474,2530,187,3

%N Symmetrical version of Graham and Riordan 1966:k=1; t(n,m,k)=(((2*k + 1)/(m + k + 1))*Binomial[n - 1 - k, m - k]*Binomial[n + k, m + k] + ((2*k + 1)/( n - m - 1 + k + 1))*Binomial[n - 1 - k, n - m - 1 - k]*Binomial[ n + k, n - m - 1 + k])

%C Row sums are:

%C {6, 18, 56, 180, 594, 2002, 6864, 23868, 83980,...}.

%C First triangle sequence in k is twice A001263 ( Narayana numbers).

%C Successive k terms shift over one so that the early terms are zero.

%C A symmetrical/ reverse coefficient/ toral transform was used to make the result a symmetrical sequence.

%D J. Riordan, Combinatorial Identities, Wiley, 1968, p.17

%D Graham,R. L. and J. Riordan,The solution of a certain recurrence, Amer. Masth. Monthly 73, 1966,pp. 604-608

%F k=1;

%F t(n,m,k)=(((2*k + 1)/(m + k + 1))*Binomial[n - 1 - k, m - k]*Binomial[n + k, m + k] +

%F ((2*k + 1)/( n - m - 1 + k + 1))*Binomial[n - 1 - k, n - m - 1 - k]*Binomial[ n + k, n - m - 1 + k])

%e {3, 3},

%e {3, 12, 3},

%e {3, 25, 25, 3},

%e {3, 42, 90, 42, 3},

%e {3, 63, 231, 231, 63, 3},

%e {3, 88, 490, 840, 490, 88, 3},

%e {3, 117, 918, 2394, 2394, 918, 117, 3},

%e {3, 150, 1575, 5796, 8820, 5796, 1575, 150, 3},

%e {3, 187, 2530, 12474, 26796, 26796, 12474, 2530, 187, 3}

%t Clear[t, n, m, k];

%t t[n_, m_, k_] = (((2*k + 1)/(m + k + 1))*Binomial[ n - 1 - k, m - k]*Binomial[n + k, m + k] + ((2*k + 1)/(n - m - 1 + k + 1))*Binomial[n - 1 - k, n - m - 1 - k]*Binomial[n + k, n - m - 1 + k]);

%t Table[Table[Table[t[n, m, k], {m, 0, n - 1}], {n, k + 1, 10}], {k, 0, 9}];

%t Table[Flatten[Table[Table[t[n, m, k], {m, 0, n - 1}], {n, k + 1, 10}]], {k, 0, 9}]

%Y A001263

%K nonn,tabl,uned

%O 0,1

%A _Roger L. Bagula_, Feb 14 2009

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Last modified July 31 10:51 EDT 2021. Contains 346373 sequences. (Running on oeis4.)