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%I #16 Jul 20 2023 10:44:15
%S 0,0,1,2,7,20,61,182,546,1632,4875,14542,43340,129064,384111,1142610,
%T 3397656,10100448,30020283,89213094,265096455,787695636,2340488535,
%U 6954401762,20664628438,61406952800,182488572045,542358944946
%N E.g.f. exp(x)*(Bessel_I(2,2*x) - Bessel_I(3,2*x) + Bessel_I(4,2*x)).
%C Third column in number triangle A116407.
%H G. C. Greubel, <a href="/A116408/b116408.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum{k=0..n} C(n,k)*(-1)^k*(C(k+1,k/2-1)*(1+(-1)^k)/2 + C(k,(k-1)/2-1)*(1-(-1)^k)/2).
%F Conjecture: 2*(n+4)*(n-23)*a(n) + 2*(n^2+64n+79)*a(n-1) +(-59n^2+232n+740)*a(n-2) +(64n^2-585n+1019)*a(n-3) +3(n-3)*(41n-124)*a(n-4)=0. - _R. J. Mathar_, Dec 10 2011
%F Shorter recurrence: (n-2)*(n+4)*(3*n^2-5*n+36)*a(n) = n*(6*n^3-n^2+70*n-139)*a(n-1) + 3*(n-1)*n*(3*n^2+n+34)*a(n-2) - _Vaclav Kotesovec_, Jun 26 2013
%F a(n) ~ 3^(n+1/2)/(2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Jun 26 2013
%t With[{nn=30},CoefficientList[Series[Exp[x](BesselI[2,2x]-BesselI[3,2x ]+ BesselI[ 4,2x]),{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Mar 13 2013 *)
%K easy,nonn
%O 0,4
%A _Paul Barry_, Feb 13 2006
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