A014205 -id:A014205 - cp's OEIS frontend

A085791 Triangle T(n,k) read by rows; given by [0,1,0,1,0,1,0,1,...] DELTA [1,0,1,1,1,2,1,3,1,4,1,5,...], where DELTA is the operator defined in A084938.

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 1, 0, 1, 10, 24, 16, 1, 0, 1, 15, 60, 86, 40, 1, 0, 1, 21, 125, 301, 312, 116, 1, 0, 1, 28, 231, 826, 1443, 1215, 395, 1, 0, 1, 36, 392, 1932, 4992, 7055, 5187, 1551, 1, 0, 1, 45, 624, 4032, 14274, 29629, 36242, 24280

Offset: 0

Philippe Deléham, Jul 23 2003

			1;
0, 1;
0, 1, 1;
0, 1, 3, 1;
0, 1, 6, 7, 1;
0, 1, 10, 24, 16, 1; ....

Cf. A084938, A085838.

Cf. A000110, A000217, A014205, A029761.

Sum_{k>=0} T(n,k) = A000110(n).
T(n + 1; 2) = A000217(n).
T(n + 2; 3) = A014205(n)= (1/12)*(n + 5)*(n + 1)*n^2.
T(n + 2; n + 1) = A029761(n).

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

Original entry on oeis.org

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

Philippe Deléham, Jul 13 2007

			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;
  ...

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.

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

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 *)

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 */

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

cp's OEIS Frontend

A085791 Triangle T(n,k) read by rows; given by [0,1,0,1,0,1,0,1,...] DELTA [1,0,1,1,1,2,1,3,1,4,1,5,...], where DELTA is the operator defined in A084938.

Views

Author

Keywords

Examples

Crossrefs

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

Formula