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 A351028 #25 Apr 25 2025 16:00:45
%S A351028 0,1,0,1,4,13,44,173,792,4009,21608,122761,737340,4696341,31665076,
%T A351028 224846037,1672266352,12976252561,104816144656,880061135057,
%U A351028 7670326372532,69286959112797,647568753568636,6251768635591613,62255057942504968,638658964709824185
%N A351028 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 2*x)) / (1 - 2*x).
%C A351028 Shifts 2 places left under 2nd-order binomial transform.
%F A351028 a(0) = 0, a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 2^k * a(n-k-2).
%F A351028 E.g.f.: BesselK(0, 1)*BesselI(0, exp(x)) - BesselI(0, 1)*BesselK(0, exp(x)). - _Ven Popov_, Apr 25 2025
%p A351028 bintr:= proc(p) local b;
%p A351028 b:= proc(n) option remember; add(p(k)*binomial(n, k), k=0..n) end
%p A351028 end:
%p A351028 b:= (bintr@@2)(a):
%p A351028 a:= n-> `if`(n<2, n, b(n-2)):
%p A351028 seq(a(n), n=0..25); # _Alois P. Heinz_, Apr 07 2025
%t A351028 nmax = 25; A[_] = 0; Do[A[x_] = x + x^2 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A351028 a[0] = 0; a[1] = 1; 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 A351028 (* another pprogram *)
%t A351028 B[x_] := BesselK[0, 1]*BesselI[0, Exp[x]] - BesselI[0, 1]*BesselK[0, Exp[x]];
%t A351028 a[n_] := SeriesCoefficient[FullSimplify[Series[B[x], {x, 0, n}]], n] n!;
%t A351028 Table[a[n], {n, 0, 30}] (* _Ven Popov_, Apr 25 2025 *)
%Y A351028 Cf. A000995, A004211, A007472, A351053, A351128, A351132, A351143, A351161.
%K A351028 nonn
%O A351028 0,5
%A A351028 _Ilya Gutkovskiy_, Feb 03 2022