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A202710 Triangle 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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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).

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

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: A134308 A209240 A263989 * A200965 A117427 A097761

Adjacent sequences:  A202707 A202708 A202709 * A202711 A202712 A202713

KEYWORD

nonn,tabl

AUTHOR

Vladimir Kruchinin, Dec 23 2011

STATUS

approved

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Last modified May 22 05:51 EDT 2022. Contains 353933 sequences. (Running on oeis4.)