A381889 Expansion of e.g.f.: (BesselI(0, 2*x) + BesselI(1, 2*x))^2*exp(2*x).

1, 4, 18, 86, 428, 2192, 11468, 60986, 328532, 1788368, 9819128, 54302712, 302157424, 1690193728, 9497996152, 53588976802, 303434431108, 1723578967056, 9818195961512, 56071829010968, 320970950634288, 1841213871449152, 10582333064327824, 60929582362628968, 351385363433883472

Offset: 0

Author

Mélika Tebni, Mar 09 2025

Keywords

• nonn

Comments

Binomial transform of A151093.

For p prime, a(p) - 2 == 0 (mod 2*p).

Crossrefs

Cf. A001405, A001700, A005566, A151093.

Programs

Maple

a := n-> add(binomial(n, k)*binomial(n-k, iquo(n-k,2))*binomial(2*k+1,k+1), k = 0 .. n): seq(a(n), n = 0 .. 24);

Mathematica

len := 24; Table[n!,{n, 0, len}] CoefficientList[Series[(BesselI[0, 2x] + BesselI[1, 2x])^2 Exp[2x], {x, 0, len}], x]  (* Peter Luschny, Mar 19 2025 *)

PARI

my(x='x+O('x^30)); Vec(serlaplace((besseli(0, 2*x) + x*besseli(1, 2*x))^2*exp(2*x))) \\ Michel Marcus, Mar 11 2025

Python

from math import comb as C
def a(n):
    return sum(C(n, k)*2**(n-k)*C(k, k//2)*C(k+1, (k+1)//2) for k in range(n+1))
print([a(n) for n in range(25)])

Formula

a(n) = Sum_{k=0..n} binomial(n, k)*2^(n-k)*A005566(k).

a(n) = Sum_{k=0..n} binomial(n, k)*A001405(n-k)*A001700(k).

a(n) = Sum_{k=0..n} binomial(n, k)*A005773(n-k+1)*A005773(k+1). - Mélika Tebni, Mar 19 2025