A190156 Expansion of (1-x-3*x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^3*(1+2*x)).

1, 1, 4, 8, 24, 61, 175, 486, 1405, 4059, 11924, 35223, 105007, 314867, 950018, 2880620, 8775638, 26843704, 82420464, 253916555, 784672011, 2431695541, 7555381574, 23531026853, 73448858179, 229730744171, 719914525210, 2260031465504, 7106721944206

Offset: 0

Author

Emanuele Munarini, May 05 2011

Keywords

• nonn

Comments

Diagonal sums of the Riordan matrix (g(x),x*g(x)), where g(x) = (1-x-x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^2) (A132276).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.

Crossrefs

Cf. A132276.

Programs

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x-3*x^2-Sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^3*(1+2*x)))); // G. C. Greubel, Oct 22 2018

Maple

T := (n, k) -> simplify(2^k*binomial(n-k, k)*hypergeom([-k, k-n-1], [2], 1/2)):
seq(add(T(n, k), k=0..floor(n/2)), n=0..28); # Peter Luschny, Oct 19 2020

Mathematica

CoefficientList[Series[(1-x-3x^2-Sqrt[1-2x-5x^2+2x^3+x^4])/(2x^3(1+2x)),{x,0,28}],x]

Maxima

a(n):=sum(sum(2^(m-j+1)*binomial(n-m,j-1)*binomial(n-m+2,j)*binomial(n-m-j+1,m-j+1),j,0,n-m+2)/(n-m+2),m,0,n); /* Vladimir Kruchinin, Oct 19 2020 */

PARI

x='x+O('x^66); Vec((1-x-3*x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^3*(1+2*x))) /* Joerg Arndt, May 15 2011 */

Formula

G.f.: (1-x-3*x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^3*(1+2*x)).

D-finite with recurrence: (n+3)*a(n) +3*a(n-1) +3*(-3*n-2)*a(n-2) +(-8*n-3)*a(n-3) +(5*n-9) *a(n-4) +2*(n-3)*a(n-5)=0. - R. J. Mathar, Oct 08 2016

a(n) = Sum_{m=0..n} Sum_{j=0..n-m+2} 2^(m-j+1)*C(n-m,j-1)*C(n-m+2,j)*C(n-m-j+1,m-j+1)/(n-m+2). - Vladimir Kruchinin, Oct 19 2020

a(n) = Sum_{k=0..floor(n/2)} 2^k*binomial(n-k, k)*hypergeom([-k, k-n-1], [2], 1/2). - Peter Luschny, Oct 19 2020