A202710 Triangle read by rows. T(n, k) = coefficient of x^n in the Taylor expansion of [((1 - x - 2*x^2 - sqrt(1 - 2*x - 3*x^2))/(2*x^2))]^k.

1, 2, 1, 4, 4, 1, 9, 12, 6, 1, 21, 34, 24, 8, 1, 51, 94, 83, 40, 10, 1, 127, 258, 267, 164, 60, 12, 1, 323, 707, 825, 604, 285, 84, 14, 1, 835, 1940, 2488, 2084, 1185, 454, 112, 16, 1, 2188, 5337, 7389, 6890, 4527, 2106, 679, 144, 18, 1, 5798, 14728, 21726, 22120, 16325, 8838, 3479, 968, 180, 20, 1

Offset

1,2

Comments

Triangle T(n,k)=

1. Riordan Array (1,((1-x-2*x^2-sqrt(1-2*x-3*x^2))/(2*x^2))) without first column.

2. Riordan Array (((1-x-2*x^2-sqrt(1-2*x-3*x^2))/(2*x)),((1-x-2*x^2-sqrt(1-2*x-3*x^2))/(2*x^2))) numbering triangle (0,0).

3. The leftmost column contains the Motzkin numbers A001006 without a(0).

The convolution triangle of the Motzkin numbers. - Peter Luschny, Oct 07 2022

Links

Table of n, a(n) for n=1..66.

Formula

T(n,k) = Sum_{i=1..k} (i*(-1)^(k-i)*binomial(k,i)*Sum_{j=floor(n/2)..n} binomial(j+i,-n+2*j)*binomial(n+i,j+i)))/(n+i)).

T(n,k) = k*Sum_{i=0..n-k} binomial(k+i,n-k-i)*binomial(n,i)/(k+i). - Vladimir Kruchinin, Dec 09 2016

Example

1,
2, 1,
4, 4, 1,
9, 12, 6, 1,
21, 34, 24, 8, 1,
51, 94, 83, 40, 10, 1,
127, 258, 267, 164, 60, 12, 1

Maple

# Uses function PMatrix from A357368. Adds a row and a column for n, k = 0.
PMatrix(10, n -> simplify(hypergeom([(1-n)/2, -n/2], [2], 4))); # Peter Luschny, Oct 06 2022

Mathematica

T[n_, k_] := Binomial[n - 1, n - k] + k*Sum[Binomial[n, i]*Binomial[k + i, n - k - i]/(k + i), {i, 0, n - k - 1}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 06 2016, after Vladimir Kruchinin *)

Prog

(Maxima) T(n, k):=sum((i*(-1)^(k-i)*binomial(k, i)*sum(binomial(j+i, -n+2*j)*binomial(n+i, j+i) , j, floor(n/2), n))/(n+i), i, 1, k);
(Maxima)
T(n, k):=+binomial(n-1, n-k)+k*sum((binomial(n, i)*binomial(k+i, n-k-i))/(k+i), i, 0, n-k-1); /* Vladimir Kruchinin, Dec 06 2016*/

Crossrefs

Cf. A001006.

Sequence in context: A113988 A209240 A263989 * A200965 A117427 A097761

Adjacent sequences: A202707 A202708 A202709 * A202711 A202712 A202713

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Dec 23 2011

Status

approved