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A156197 T(n,k) = A009766(n,k) + A009766(n,n-k), triangle read by rows.

%I A156197 #22 Oct 05 2024 16:31:21
%S A156197 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 A156197 110,138,133,430,436,324,240,240,324,436,430,1431,1438,1036,682,550,
%U A156197 682,1036,1438,1431,4863,4871,3476,2156,1430,1430,2156,3476,4871,4863
%N A156197 T(n,k) = A009766(n,k) + A009766(n,n-k), triangle read by rows.
%H A156197 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 A156197 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 A156197 From _Roger L. Bagula_ and _Gary W. Adamson_, Dec 03 2009: (Start)
%F A156197 T(n,k) = ((n - k + 1)*binomial(n + k, n) + (k + 1)*binomial(-k + 2*n, n))/(n + 1).
%F A156197 T(n,k) = A009766(n,k) + A033184(n,k). (End)
%F A156197 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 A156197 Triangle begins:
%e A156197       2;
%e A156197       2,    2;
%e A156197       3,    4,    3;
%e A156197       6,    8,    8,    6;
%e A156197      15,   18,   18,   18,   15;
%e A156197      43,   47,   42,   42,   47,   43;
%e A156197     133,  138,  110,   96,  110,  138,  133;
%e A156197     430,  436,  324,  240,  240,  324,  436,  430;
%e A156197    1431, 1438, 1036,  682,  550,  682, 1036, 1438, 1431;
%e A156197    4863, 4871, 3476, 2156, 1430, 1430, 2156, 3476, 4871, 4863;
%e A156197     ...
%t A156197 t0[n_, m_] = Binomial[n + m, n] - Binomial[n + m, m - 1];
%t A156197 T[n_, m_] = FullSimplify[t0[n, m] + t0[n, n - m]];
%t A156197 Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}] // Flatten
%t A156197 (* or *)
%t A156197 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 A156197 (Maxima) A009766(n, k) := binomial(n + k, n)*(n - k + 1)/(n + 1)$
%o A156197 create_list(A009766(n, k) + A009766(n, n - k), n, 0, 10, k, 0, n); /* _Franck Maminirina Ramaharo_, Dec 11 2018 */
%Y A156197 Row sums: 2*A000108(n+1).
%Y A156197 Cf. A009766, A033184. - _Roger L. Bagula_ and _Gary W. Adamson_, Dec 03 2009.
%K A156197 nonn,tabl,easy
%O A156197 0,1
%A A156197 _Roger L. Bagula_, Feb 05 2009
%E A156197 Edited by _Franck Maminirina Ramaharo_, Dec 11 2018