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
Examples
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; ...
Links
- 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.
Programs
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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 *) -
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 */
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