A116400 Expansion of e.g.f. Bessel_I(2,2x) + Bessel_I(3,2x) + Bessel_I(4,2x).

0, 0, 1, 1, 5, 5, 21, 21, 84, 84, 330, 330, 1287, 1287, 5005, 5005, 19448, 19448, 75582, 75582, 293930, 293930, 1144066, 1144066, 4457400, 4457400, 17383860, 17383860, 67863915, 67863915, 265182525, 265182525, 1037158320, 1037158320, 4059928950, 4059928950, 15905368710

Offset

0,5

Comments

T(n,2) in number triangle A116399.

Double of A002054.

Links

Table of n, a(n) for n=0..36.

Formula

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

a(n) = C(n+1,n/2-1)(1+(-1)^n)/2 + C(n,(n-1)/2-1)(1-(-1)^n)/2.

a(2n) = a(2n+1) = A002054(n).

a(n) ~ c * 2^(n+1/2) / sqrt(Pi*n), where c = 1 if n is odd and c = 2 if n is even. - Amiram Eldar, Sep 23 2025

Mathematica

a[n_] := Binomial[2 * Floor[n/2] + 1, Floor[n/2] - 1]; Array[a, 30, 0] (* Amiram Eldar, Sep 23 2025 *)

Prog

(PARI) a(n)=n=n\2; binomial(2*n+1, n-1);

Crossrefs

Cf. A000108, A002054, A116399.

Sequence in context: A171219 A325893 A146043 * A279810 A279750 A052992

Adjacent sequences: A116397 A116398 A116399 * A116401 A116402 A116403

Keyword

easy,nonn

Author

Paul Barry, Feb 13 2006

Extensions

More terms from Amiram Eldar, Sep 23 2025

Status

approved