cp's OEIS Frontend

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

%I A116385 #35 Jan 23 2025 10:23:09
%S A116385 0,0,1,2,5,10,21,42,84,168,330,660,1287,2574,5005,10010,19448,38896,
%T A116385 75582,151164,293930,587860,1144066,2288132,4457400,8914800,17383860,
%U A116385 34767720,67863915,135727830,265182525,530365050,1037158320,2074316640
%N A116385 Expansion of e.g.f. Bessel_I(2,2x) + 2*Bessel_I(3,2x) + Bessel_I(4,2x).
%C A116385 Third column of the Riordan array A116382.
%C A116385 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
%H A116385 Reinhard Zumkeller, <a href="/A116385/b116385.txt">Table of n, a(n) for n = 0..1000</a>
%H A116385 Gennady Eremin, <a href="https://arxiv.org/abs/2211.01135">Dyck Numbers, II. Triplets and Rooted Trees in OEIS A036991</a>, arXiv:2211.01135 [math.NT], 2022.
%F A116385 E.g.f.: (d/dx)(Bessel_I(3,2x),x) + 2*Bessel_I(3,2x).
%F A116385 a(n) = C(n+1,floor((n-2)/2))*(1+(-1)^n)/2 + C(n,floor((n-3)/2))*(1-(-1)^n).
%F A116385 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
%F A116385 a(n) = A001405(n+3) - 3*A001405(n+1) (from Eremin link). - _Bill McEachen_, Dec 12 2022
%F A116385 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
%t A116385 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 *)
%o A116385 (Haskell)
%o A116385 a116385 n = a051631 (n+1) $ (n+1) `div` 2
%o A116385 -- _Reinhard Zumkeller_, Nov 13 2011
%o A116385 (PARI) a(n)= binomial(n+3, (n+3)\2) - 3*binomial(n+1, (n+1)\2) \\ _Bill McEachen_, Dec 12 2022
%Y A116385 Cf. A001405.
%K A116385 easy,nonn
%O A116385 0,4
%A A116385 _Paul Barry_, Feb 12 2006