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

Triangle begins:
  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