A130749 - OEIS

%I #25 Jan 08 2022 22:02:34

%S 1,1,1,1,3,1,1,7,6,1,1,15,24,10,1,1,31,80,60,15,1,1,63,240,280,125,21,

%T 1,1,127,672,1120,770,231,28,1,1,255,1792,4032,3920,1806,392,36,1,1,

%U 511,4608,13440,17472,11340,3780,624,45,1

%N Triangle A007318*A090181 (as infinite lower triangular matrices) .

%H Paul Barry, <a href="https://arxiv.org/abs/1807.05794">Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences</a>, arXiv:1807.05794 [math.CO], 2018.

%H Sherry H. F. Yan, <a href="http://arXiv.org/abs/0805.2465">Schroeder Paths and Pattern Avoiding Partitions</a>, arXiv:0805.2465 [math.CO], 2008-2009; Corollary 3.6.

%F Sum_{k=0..n} T(n,k) = A007317(n+1).

%F G.f.: 1/(1-x-xy/(1-x/(1-x-xy/(1-x/(1-x-xy/(1-x.... (continued fraction); [_Paul Barry_, Jan 12 2009]

%F T(n,k) = Sum_{i=1..n} binomial(n, i)*N(i,k), T(n,0)=1, where N(n,k) is the triangle of Narayana numbers A001263. - _Vladimir Kruchinin_, Jan 08 2022

%e Triangle begins:

%e   1;

%e   1,   1;

%e   1,   3,    1;

%e   1,   7,    6,     1;

%e   1,  15,   24,    10,     1;

%e   1,  31,   80,    60,    15,     1;

%e   1,  63,  240,   280,   125,    21,    1;

%e   1, 127,  672,  1120,   770,   231,   28,   1;

%e   1, 255, 1792,  4032,  3920,  1806,  392,  36,  1;

%e   1, 511, 4608, 13440, 17472, 11340, 3780, 624, 45,  1;

%e   ...

%t nmax = 9;

%t T1[n_, k_] := Binomial[n, k];

%t T2[n_, k_] := Sum[(-1)^(j-k) Binomial[2n-j, j] Binomial[j, k] CatalanNumber[n-j], {j, 0, n}];

%t T[n_, k_] := Sum[T1[n, m] T2[m, k], {m, 0, n}];

%t Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 10 2018 *)

%o (Maxima)

%o N(n, k):=(binomial(n, k-1)*binomial(n, k))/n;

%o T(n, k):=if k=0 then 1 else sum(binomial(n, i)*N(i, k), i, 1, n); /* _Vladimir Kruchinin_, Jan 08 2022 */

%Y Cf. A000012, A000225, A001788, A003472 ; A000012, A000217, A014205.

%K nonn,tabl

%O 0,5

%A _Philippe Deléham_, Jul 13 2007