This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A263986 #26 Apr 12 2023 11:06:55
%S A263986 1,0,1,3,11,37,124,416,1409,4828,16741,58697,207868,742667,2674063,
%T A263986 9694235,35356683,129643193,477636116,1767259009,6564113655,
%U A263986 24466256074,91482545929,343059584993,1289904100956,4861946326427,18367352950759,69533550719586
%N A263986 Difference between Catalan numbers and Fibonacci numbers: a(n) = C(n) - F(n).
%H A263986 Robert Israel, <a href="/A263986/b263986.txt">Table of n, a(n) for n = 0..1667</a>
%H A263986 S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt">Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017)
%F A263986 a(n) = (2*n)!/(n!*(n+1)!) - ((1+sqrt(5))^n-(1-sqrt(5))^n)/(2^n*sqrt(5)).
%F A263986 a(n) = A000108(n) - A000045(n).
%F A263986 G.f.: (1 - sqrt(1 - 4*x))/(2*x) - x/(1 - x - x^2).
%F A263986 a(n+1)/a(n) ~ 4.
%F A263986 Conjecture: (11*n-20)*(n-3)*(n+1)*a(n) +(-55*n^3+276*n^2-389*n+120)*a(n-1) +3*(11*n^3-64*n^2+125*n-80)*a(n-2) +2*(2*n-5)*(n-2)*(11*n-9)*a(n-3)=0. - _R. J. Mathar_, Jun 07 2016
%F A263986 Conjecture: +(n+1)*a(n) +2*(-3*n+1)*a(n-1) +(7*n-11)*a(n-2) +2*(3*n-7)*a(n-3) +(-7*n+25)*a(n-4) +2*(-2*n+9)*a(n-5)=0. - _R. J. Mathar_, Jun 07 2016
%F A263986 Conjectures confirmed since the g.f. satisfies the differential equations corresponding to these recurrences. - _Robert Israel_, Apr 26 2017
%F A263986 E.g.f.: (BesselI(0,2*x) - BesselI(1,2*x))*exp(2*x) - 2*sinh(sqrt(5)*x/2)*exp(x/2)/sqrt(5). - _Ilya Gutkovskiy_, Jun 07 2016
%p A263986 f:= gfun:-rectoproc({(2+4*n)*a(n)+(10+7*n)*a(1+n)+(-16-6*n)*a(n+2)+(-24-7*n)*a(n+3)+(28+6*n)*a(n+4)+(-6-n)*a(n+5), a(0) = 1, a(1) = 0, a(2) = 1, a(3) = 3, a(4) = 11}, a(n), remember):
%p A263986 map(f, [$0..40]); # _Robert Israel_, Apr 26 2017
%t A263986 RecurrenceTable[{(2+4n) a[n] + (10+7n) a[1+n] + (-16-6n) a[n+2] + (-24-7n) a[n+3] + (28+6n) a[n+4] + (-6-n) a[n+5] == 0, a[0] == 1, a[1] == 0, a[2] == 1, a[3] == 3, a[4] == 11}, a, {n, 0, 40}] (* _Jean-François Alcover_, Sep 16 2022, after _Robert Israel_ *)
%o A263986 (PARI) a000108(n) = binomial(2*n, n)/(n+1)
%o A263986 a(n) = a000108(n)-fibonacci(n) \\ _Felix Fröhlich_, Jun 07 2016
%Y A263986 Cf. A000108, A000045.
%K A263986 nonn
%O A263986 0,4
%A A263986 _Ran Pan_, Oct 31 2015