A130749 Triangle A007318*A090181 (as infinite lower triangular matrices) .

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, 1, 1, 127, 672, 1120, 770, 231, 28, 1, 1, 255, 1792, 4032, 3920, 1806, 392, 36, 1, 1, 511, 4608, 13440, 17472, 11340, 3780, 624, 45, 1

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.

Sherry H. F. Yan, Schroeder Paths and Pattern Avoiding Partitions, arXiv:0805.2465 [math.CO], 2008-2009; Corollary 3.6.

Formula

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

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]

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

Example

Triangle begins:
  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,    1;
  1, 127,  672,  1120,   770,   231,   28,   1;
  1, 255, 1792,  4032,  3920,  1806,  392,  36,  1;
  1, 511, 4608, 13440, 17472, 11340, 3780, 624, 45,  1;
  ...

Mathematica

nmax = 9;
T1[n_, k_] := Binomial[n, k];
T2[n_, k_] := Sum[(-1)^(j-k) Binomial[2n-j, j] Binomial[j, k] CatalanNumber[n-j], {j, 0, n}];
T[n_, k_] := Sum[T1[n, m] T2[m, k], {m, 0, n}];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

Prog

(Maxima)
N(n, k):=(binomial(n, k-1)*binomial(n, k))/n;
T(n, k):=if k=0 then 1 else sum(binomial(n, i)*N(i, k), i, 1, n); /* Vladimir Kruchinin, Jan 08 2022 */

Crossrefs

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

Sequence in context: A154341 A348863 A202181 * A250118 A250119 A154959

Adjacent sequences: A130746 A130747 A130748 * A130750 A130751 A130752

Keyword

nonn,tabl

Author

Philippe Deléham, Jul 13 2007

Status

approved