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 A351143 #16 Apr 25 2025 16:00:49
%S A351143 1,0,1,2,5,16,61,258,1177,5776,30537,173394,1050045,6732608,45459493,
%T A351143 322141106,2390075249,18525967328,149684238801,1257802518754,
%U A351143 10969260208565,99100423076912,926030783479629,8937741026924450,88988433270106249,912906193294355952
%N A351143 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 2*x)) / (1 - 2*x).
%C A351143 Shifts 2 places left under 2nd-order binomial transform.
%F A351143 a(0) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 2^k * a(n-k-2).
%F A351143 E.g.f.: BesselK(1, 1)*BesselI(0, exp(x)) + BesselI(1, 1)*BesselK(0, exp(x)). - _Ven Popov_, Apr 25 2025
%p A351143 bintr:= proc(p) local b;
%p A351143 b:= proc(n) option remember; add(p(k)*binomial(n, k), k=0..n) end
%p A351143 end:
%p A351143 b:= (bintr@@2)(a):
%p A351143 a:= n-> `if`(n<2, 1-n, b(n-2)):
%p A351143 seq(a(n), n=0..25); # _Alois P. Heinz_, Apr 07 2025
%t A351143 nmax = 25; A[_] = 0; Do[A[x_] = 1 + x^2 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A351143 a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 2^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
%t A351143 (* another program *)
%t A351143 B[x_] := BesselK[1, 1]*BesselI[0, Exp[x]] + BesselI[1, 1]*BesselK[0, Exp[x]];
%t A351143 a[n_] := SeriesCoefficient[FullSimplify[Series[B[x], {x, 0, n}]], n] n!
%t A351143 Table[a[n], {n, 0, 30}] (* _Ven Popov_, Apr 25 2025 *)
%Y A351143 Cf. A000994, A004211, A007472, A351144, A351150, A351151, A351152.
%K A351143 nonn
%O A351143 0,4
%A A351143 _Ilya Gutkovskiy_, Feb 02 2022