A106188 Expansion of 1/((1-x^2)*sqrt(1-4*x)).

1, 2, 7, 22, 77, 274, 1001, 3706, 13871, 52326, 198627, 757758, 2902783, 11158358, 43019383, 166275878, 644099773, 2499882098, 9719235073, 37845145898, 147565763893, 576103020338, 2251664727613, 8809533747938, 34499268410713

Offset: 0

Author

Paul Barry, Apr 24 2005

Keywords

• easy
• nonn

Comments

Diagonal sums of number triangle A106187.

Examples

			1 + 2*x + 7*x^2 + 22*x^3 + 77*x^4 + 274*x^5 + 1001*x^6 + 3706*x^7 + 13871*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A006134, A054108. Convolution of A000984 and A059841.

Programs

Mathematica

CoefficientList[Series[1/((1-x^2)*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 03 2014 *)

PARI

x='x+O('x^50); Vec(1/((1-x^2)*sqrt(1-4*x))) \\ G. C. Greubel, Mar 16 2017

Formula

G.f. 1 / ((1 - x^2) * sqrt(1 - 4*x)).

a(n)=sum{k=0..floor(n/2), binomial(2(n-2k), n-2k)}.

PSUMSIGN transform of A006134. a(n+1) + a(n) = A006134(n). a(n) = Sum_{k=0..n} (-1)^k * binomial(2 * (n-k), n-k). - Michael Somos, Jun 20 2012

First difference is A054108. a(n+1) - a(n) = A054108(n). - Michael Somos, Jun 20 2012

D-finite with recurrence: n*a(n)+2*(1-2*n)*a(n-1) -n*a(n-2) +2*(2*n-1)*a(n-3)=0. - R. J. Mathar, Nov 09 2012

a(n) ~ 2^(2*n+4) / (15*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 03 2014