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A105868 Triangle read by rows, T(n,k) = C(n,k)*C(k,n-k). 5
1, 0, 1, 0, 2, 1, 0, 0, 6, 1, 0, 0, 6, 12, 1, 0, 0, 0, 30, 20, 1, 0, 0, 0, 20, 90, 30, 1, 0, 0, 0, 0, 140, 210, 42, 1, 0, 0, 0, 0, 70, 560, 420, 56, 1, 0, 0, 0, 0, 0, 630, 1680, 756, 72, 1, 0, 0, 0, 0, 0, 252, 3150, 4200, 1260, 90, 1, 0, 0, 0, 0, 0, 0, 2772, 11550, 9240, 1980, 110, 1, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are the central trinomial coefficients A002426.

Product of A007318 and this sequence is A008459.

Coefficient array for polynomials P(n,x) = x^n*F(1/2-n/2,-n/2;1;4/x). - Paul Barry, Oct 04 2008

Column sums give A001850. It appears that the sums along the antidiagonals of the triangle produce A182883. - Peter Bala, Mar 06 2013

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows

FORMULA

G.f.: 1/(sqrt((1-x*y)^2-4*x^2*y)). - Vladimir Kruchinin, Oct 28 2020

EXAMPLE

Triangle begins

1;

0, 1;

0, 2, 1;

0, 0, 6, 1;

0, 0, 6, 12, 1;

0, 0, 0, 30, 20, 1;

MAPLE

gf := 1/((1 - x*y)^2 - 4*y^2*x)^(1/2):

yser := series(gf, y, 12): ycoeff := n -> coeff(yser, y, n):

row := n -> seq(coeff(expand(ycoeff(n)), x, k), k=0..n):

seq(row(n), n=0..7); # Peter Luschny, Oct 28 2020

MATHEMATICA

Flatten[Table[Binomial[n, k]Binomial[k, n-k], {n, 0, 20}, {k, 0, n}]] (* Harvey P. Dale, Nov 12 2014 *)

PROG

(Magma) [[Binomial(n, k)*Binomial(k, n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jun 14 2015

CROSSREFS

Cf. A063007. A001850 (column sums), A182883.

Sequence in context: A057150 A185663 A262125 * A267163 A357885 A265163

Adjacent sequences: A105865 A105866 A105867 * A105869 A105870 A105871

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Apr 23 2005

STATUS

approved

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Last modified November 30 04:37 EST 2022. Contains 358431 sequences. (Running on oeis4.)