A330796 a(n) = Sum_{k=0..n} binomial(n, k)*(2^k - binomial(k, floor(k/2))).

0, 1, 4, 14, 46, 147, 462, 1437, 4438, 13637, 41746, 127426, 388076, 1179739, 3581052, 10856790, 32880942, 99496293, 300845658, 909073356, 2745419352, 8287110075, 25003877784, 75412396575, 227366950140, 685293578217, 2064924137152, 6220442229932, 18734334462598

Offset: 0

Author

Peter Luschny, Jan 12 2020

Keywords

• nonn

Links

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

Crossrefs

Cf. A008549, A045621.

Programs

Magma

m:=40;
R:=PowerSeriesRing(Rationals(), m+2);
A330796:= func< n | Coefficient(R!( (1-x-Sqrt(1-3*x)*Sqrt(1+x))/(2*x*(1-3*x)) ), n) >;
[A330796(n): n in [0..m]]; // G. C. Greubel, Sep 14 2023

Maple

gf := exp(x)*(exp(2*x) - BesselI(0,2*x) - BesselI(1,2*x)):
ser := series(gf, x, 32): seq(n!*coeff(ser, x, n), n=0..28);
# Alternative:
a := proc(n) option remember; if n < 3 then return n^2 fi;
((18-9*n)*a(n-3) - (3*n+3)*a(n-2) + (5*n+2)*a(n-1))/(n+1) end:
seq(a(n), n=0..28);
# Or:
a := n -> add(binomial(n, k)*(2^k - binomial(k, floor(k/2))), k=0..n):
seq(a(n), n=0..28);

Mathematica

a[n_]:= Sum[k Binomial[n,k] Hypergeometric2F1[(k-n)/2, (k-n+1)/2, k+2, 4], {k,0,n}]; Table[a[n], {n,0,30}] (* Peter Luschny, May 24 2021 *)
(* Second program *)
A330796[n_]:= Coefficient[Series[(1-x-Sqrt[1-3*x]*Sqrt[1+x])/(2*x*(1- 3*x)), {x,0,50}], x, n];
Table[A330796[n], {n,0,30}] (* G. C. Greubel, Sep 14 2023 *)

Maxima

a(n):=(2*sum((-1)^j*binomial(2*j+1,j)*3^(n-j-1)*binomial(n+1,j+2),j,0,n-1))/(n+1); /* Vladimir Kruchinin, Sep 30 2020 */

SageMath

m=40
P. = PowerSeriesRing(ZZ, m+2)
def A330796(n): return P( (1-x-sqrt(1-3*x)*sqrt(1+x))/(2*x*(1-3*x)) ).list()[n]
[A330796(n) for n in range(m+1)] # G. C. Greubel, Sep 14 2023

Formula

a(n) = n! * [x^n] e^x*(e^(2*x) - I_{0}(2*x) - I_{1}(2*x)), where I_{n}(x) are the modified Bessel functions of the first kind.

a(n) = [x^n] (1 - x - sqrt(1 - 3*x)*sqrt(1 + x))/(2*x*(1 - 3*x)).

D-finite with recurrence a(n) = ((18-9*n)*a(n-3) - (3*n+3)*a(n-2) + (5*n+2)*a(n-1))/(n+1).

Sum_{k=0..n} binomial(n, k)*a(k) = A008549(n).

Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(k) = A045621(n).

a(n) = 2*Sum_{j=0..n-1} (-1)^j*C(2*j+1,j)*3^(n-j-1)*C(n+1,j+2)/(n+1). - Vladimir Kruchinin, Sep 30 2020

a(n) = Sum_{k=0..n} k*binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4). - Peter Luschny, May 24 2021

a(n) ~ 3^n * (1 - sqrt(3/(Pi*n))). - Vaclav Kotesovec, May 24 2021