A116385 Expansion of e.g.f. Bessel_I(2,2x) + 2*Bessel_I(3,2x) + Bessel_I(4,2x).

0, 0, 1, 2, 5, 10, 21, 42, 84, 168, 330, 660, 1287, 2574, 5005, 10010, 19448, 38896, 75582, 151164, 293930, 587860, 1144066, 2288132, 4457400, 8914800, 17383860, 34767720, 67863915, 135727830, 265182525, 530365050, 1037158320, 2074316640

Offset

0,4

Comments

Third column of the Riordan array A116382.

Apart from its root term -1: central terms of the triangle in A051631: a(n) = A051631(n+1, [(n+1)/2]). - Reinhard Zumkeller, Nov 13 2011

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Gennady Eremin, Dyck Numbers, II. Triplets and Rooted Trees in OEIS A036991, arXiv:2211.01135 [math.NT], 2022.

Formula

E.g.f.: (d/dx)(Bessel_I(3,2x),x) + 2*Bessel_I(3,2x).

a(n) = C(n+1,floor((n-2)/2))*(1+(-1)^n)/2 + C(n,floor((n-3)/2))*(1-(-1)^n).

Conjecture: (n+4)*a(n) -2*a(n-1) +(-7*n-8)*a(n-2) +6*a(n-3) +12*(n-2)*a(n-4)=0. - R. J. Mathar, Jun 13 2014

a(n) = A001405(n+3) - 3*A001405(n+1) (from Eremin link). - Bill McEachen, Dec 12 2022

G.f.: (-1 - x + x^2 + B(x) - 3*x^2*B(x))/x^3, where B(x) is the g.f. of A001405. - Gennady Eremin, Oct 09 2023

Mathematica

With[{nn=40}, CoefficientList[Series[BesselI[2, 2x]+2BesselI[3, 2x]+ BesselI[ 4, 2x], {x, 0, nn}], x]Range[0, nn]!] (* Harvey P. Dale, Sep 14 2011 *)

Prog

(Haskell)
a116385 n = a051631 (n+1) $ (n+1) `div` 2
-- Reinhard Zumkeller, Nov 13 2011
(PARI) a(n)= binomial(n+3, (n+3)\2) - 3*binomial(n+1, (n+1)\2) \\ Bill McEachen, Dec 12 2022

Crossrefs

Cf. A001405.

Sequence in context: A261681 A333798 A030525 * A267515 A215411 A243988

Adjacent sequences: A116382 A116383 A116384 * A116386 A116387 A116388

Keyword

easy,nonn

Author

Paul Barry, Feb 12 2006

Status

approved