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T(n,k) = A009766(n,k) + A009766(n,n-k), triangle read by rows.
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%I #22 Oct 05 2024 16:31:21

%S 2,2,2,3,4,3,6,8,8,6,15,18,18,18,15,43,47,42,42,47,43,133,138,110,96,

%T 110,138,133,430,436,324,240,240,324,436,430,1431,1438,1036,682,550,

%U 682,1036,1438,1431,4863,4871,3476,2156,1430,1430,2156,3476,4871,4863

%N T(n,k) = A009766(n,k) + A009766(n,n-k), triangle read by rows.

%H L. Carlitz and J. Riordan, <a href="https://dx.doi.org/10.1215/S0012-7094-64-03136-92">Two element lattice permutation numbers and their q-generalization</a>, Duke Math. J. Volume 31, Number 3 (1964), 371-388.

%F T(n,k) = -binomial(k + n, -1 + k) + binomial(k + n, n) + binomial(-k + 2*n, n) - binomial(-k + 2*n, -1 - k + n).

%F From _Roger L. Bagula_ and _Gary W. Adamson_, Dec 03 2009: (Start)

%F T(n,k) = ((n - k + 1)*binomial(n + k, n) + (k + 1)*binomial(-k + 2*n, n))/(n + 1).

%F T(n,k) = A009766(n,k) + A033184(n,k). (End)

%F G.f.: (C(t*x) + C(x)*(1 - x*C(t*x) - t*x*C(t*x)))/((1 - t*x*C(x))*(1 - x*C(t*x))), where C(x) = (1 - sqrt(1 - 4*x))/(2*x). - _Franck Maminirina Ramaharo_, Dec 11 2018

%e Triangle begins:

%e 2;

%e 2, 2;

%e 3, 4, 3;

%e 6, 8, 8, 6;

%e 15, 18, 18, 18, 15;

%e 43, 47, 42, 42, 47, 43;

%e 133, 138, 110, 96, 110, 138, 133;

%e 430, 436, 324, 240, 240, 324, 436, 430;

%e 1431, 1438, 1036, 682, 550, 682, 1036, 1438, 1431;

%e 4863, 4871, 3476, 2156, 1430, 1430, 2156, 3476, 4871, 4863;

%e ...

%t t0[n_, m_] = Binomial[n + m, n] - Binomial[n + m, m - 1];

%t T[n_, m_] = FullSimplify[t0[n, m] + t0[n, n - m]];

%t Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}] // Flatten

%t (* or *)

%t Table[Table[((1 - k + n)*Binomial[k + n, n] + (1 + k)*Binomial[-k + 2*n, n])/(1 + n), {k, 0, n}], {n, 0, 10}] // Flatten (* _Roger L. Bagula_ and _Gary W. Adamson_, Dec 03 2009 *)

%o (Maxima) A009766(n, k) := binomial(n + k, n)*(n - k + 1)/(n + 1)$

%o create_list(A009766(n, k) + A009766(n, n - k), n, 0, 10, k, 0, n); /* _Franck Maminirina Ramaharo_, Dec 11 2018 */

%Y Row sums: 2*A000108(n+1).

%Y Cf. A009766, A033184. - _Roger L. Bagula_ and _Gary W. Adamson_, Dec 03 2009.

%K nonn,tabl,easy

%O 0,1

%A _Roger L. Bagula_, Feb 05 2009

%E Edited by _Franck Maminirina Ramaharo_, Dec 11 2018