A360149 a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k,n-2*k).

1, 2, 7, 27, 107, 429, 1731, 7012, 28478, 115864, 471991, 1924483, 7852083, 32053208, 130893949, 534673600, 2184482707, 8926392419, 36479840422, 149095843951, 609400587426, 2490900041118, 10181669553847, 41618414303969, 170118507902985, 695366323719302

Offset

0,2

Links

Table of n, a(n) for n=0..25.

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^2 * c(x)^5) ), where c(x) is the g.f. of A000108.

a(n) ~ sqrt((7 - 5*(2/(173 + 21*sqrt(69)))^(1/3) + ((173 + 21*sqrt(69))/2)^(1/3)) / 69) / ((4 - (2/(25 - 3*sqrt(69)))^(1/3) - ((25 - 3*sqrt(69))/2)^(1/3))/3)^n. - Vaclav Kotesovec, Jan 28 2023

D-finite with recurrence n*(47*n-1011)*a(n) +(-261*n^2 +8567*n -6378)*a(n-1) +2*(-165*n^2 -9388*n +16143)*a(n-2) +(3089*n^2 +919*n -27492)*a(n-3) +2*(-1283*n^2 +3900*n +3981)*a(n-4) +4*(81*n+11)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Mar 02 2023

Maple

A360149 := proc(n)
    add(binomial(2*n+k, n-2*k), k=0..floor(n/2)) ;
end proc:
seq(A360149(n), n=0..40) ; # R. J. Mathar, Mar 02 2023

Mathematica

a[n_] := Sum[Binomial[2*n + k, n - 2*k], {k, 0, Floor[n/2]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)

Prog

(PARI) a(n) = sum(k=0, n\2, binomial(2*n+k, n-2*k));
(PARI) my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^2*(2/(1+sqrt(1-4*x)))^5)))

Crossrefs

Cf. A014300, A106188, A108081, A114121, A176287.

Cf. A000108, A360144, A360150.

Sequence in context: A150593 A024429 A136412 * A192417 A150594 A150595

Adjacent sequences: A360146 A360147 A360148 * A360150 A360151 A360152

Keyword

nonn

Author

Seiichi Manyama, Jan 28 2023

Status

approved