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A236934 Triangle of Poupard numbers g_n(k) read by rows, n>=1, 1<=k<=2n-1.

%I A236934 #19 Oct 14 2022 06:58:36
%S A236934 1,0,2,0,0,4,8,4,0,0,32,64,80,64,32,0,0,544,1088,1504,1664,1504,1088,
%T A236934 544,0,0,15872,31744,45440,54784,58112,54784,45440,31744,15872,0,0,
%U A236934 707584,1415168,2059264,2576384,2911744,3027968,2911744,2576384,2059264,1415168,707584,0
%N A236934 Triangle of Poupard numbers g_n(k) read by rows, n>=1, 1<=k<=2n-1.
%H A236934 Peter Luschny, <a href="/A236934/b236934.txt">Row(n) for n = 1..25</a>
%H A236934 Dominique Foata and Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~foata/paper/pub123Seidel.pdf">Seidel Triangle Sequences and Bi-Entringer Numbers</a>, November 20, 2013.
%H A236934 Foata, Dominique; Han, Guo-Niu; Strehl, Volker <a href="https://doi.org/10.1016/j.laa.2016.09.016">The Entringer-Poupard matrix sequence</a>.  Linear Algebra Appl. 512, 71-96 (2017).
%H A236934 Christiane Poupard, <a href="http://dx.doi.org/10.1016/S0195-6698(89)80009-5">Deux propriétés des arbres binaires ordonnés stricts</a>, Europ. J. Combin., vol. 10, 1989, p. 369-374.
%F A236934 4^(-n)*sum(k=1..2*n+1, binomial(2*n,k-1)*T(n+1,k)) = A000364(n), n>=0. - _Peter Luschny_, May 11 2014
%F A236934 (-1)^n*sum(k=1..2*n+1, (-1)^(k-1)*binomial(2*n,k-1)*T(n+1,k)) = A000302(n), n>=0. - _Peter Luschny_, May 11 2014
%e A236934 Triangle begins:
%e A236934 1,
%e A236934 0, 2, 0,
%e A236934 0, 4, 8, 4, 0,
%e A236934 0, 32, 64, 80, 64, 32, 0,
%e A236934 0, 544, 1088, 1504, 1664, 1504, 1088, 544, 0,
%e A236934 ...
%p A236934 T := proc(n,k) option remember; local j;
%p A236934   if n = 1 then 1
%p A236934 elif k = 1 then 0
%p A236934 elif k = 2 then 2*add(T(n-1, j), j=1..2*n-3)
%p A236934 elif k > n then T(n, 2*n-k)
%p A236934 else 2*T(n, k-1)-T(n, k-2)-4*T(n-1, k-2)
%p A236934   fi end:
%p A236934 seq(print(seq(T(n,k), k=1..2*n-1)), n=1..6); # _Peter Luschny_, May 11 2014
%t A236934 T[n_, k_] /; 1 <= k <= 2n-1 := T[n, k] = Which[n == 1, 1, k == 1, 0, k == 2, 2 Sum[T[n-1, j], {j, 1, 2n-3}], k > n, T[n, 2n-k], True, 2 T[n, k-1] - T[n, k-2] - 4 T[n-1, k-2]]; T[_, _] = 0;
%t A236934 Table[T[n, k], {n, 1, 7}, {k, 1, 2n-1}] // Flatten (* _Jean-François Alcover_, Jul 08 2019, from Maple *)
%Y A236934 Cf. A000182 (row sums), A008282, A125053.
%K A236934 nonn,tabf
%O A236934 1,3
%A A236934 _N. J. A. Sloane_, Feb 17 2014
%E A236934 More terms from _Peter Luschny_, May 11 2014