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A119326 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,2j)*C(n-k,2j).

%I A119326 #21 Mar 30 2023 16:08:30
%S A119326 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,4,4,1,1,1,1,7,10,7,1,1,1,1,11,19,
%T A119326 19,11,1,1,1,1,16,31,38,31,16,1,1,1,1,22,46,66,66,46,22,1,1,1,1,29,64,
%U A119326 106,126,106,64,29,1,1,1,1,37,85,162,226,226,162,85,37,1,1
%N A119326 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,2j)*C(n-k,2j).
%C A119326 Third column is essentially A000124. Fourth column is essentially A005448. Fifth column is A119327. Product of Pascal's triangle A007318 and A119328. Row sums are A038504. T(n,k) = T(n,n-k).
%D A119326 Lukas Spiegelhofer and Jeffrey Shallit, Continuants, Run Lengths, and Barry's Modified Pascal Triangle, Volume 26(1) 2019, of The Electronic Journal of Combinatorics, #P1.31.
%H A119326 Seiichi Manyama, <a href="/A119326/b119326.txt">Rows n = 0..139, flattened</a>
%H A119326 Jeffrey Shallit, Lukas Spiegelhofer, <a href="https://arxiv.org/abs/1710.06203">Continuants, run lengths, and Barry's modified Pascal triangle</a>, arXiv:1710.06203 [math.CO], 2017.
%F A119326 Column k has g.f.: (x^k/(1-x))* Sum{j=0..k} C(k,2j)*(x/(1-x))^(2j).
%F A119326 T(2n,n) = A119358(n). - _Alois P. Heinz_, Aug 31 2018
%e A119326 Triangle begins:
%e A119326   1;
%e A119326   1, 1;
%e A119326   1, 1,  1;
%e A119326   1, 1,  1,  1;
%e A119326   1, 1,  2,  1,  1;
%e A119326   1, 1,  4,  4,  1,  1;
%e A119326   1, 1,  7, 10,  7,  1, 1;
%e A119326   1, 1, 11, 19, 19, 11, 1, 1;
%e A119326   ...
%Y A119326 Cf. A119358.
%K A119326 easy,nonn,tabl
%O A119326 0,13
%A A119326 _Paul Barry_, May 14 2006