OFFSET
0,3
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.
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
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]
PROG
(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 */
(Magma) m:=50; R<x>:=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
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Emanuele Munarini, May 05 2011
STATUS
approved