A351028 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 2*x)) / (1 - 2*x).

0, 1, 0, 1, 4, 13, 44, 173, 792, 4009, 21608, 122761, 737340, 4696341, 31665076, 224846037, 1672266352, 12976252561, 104816144656, 880061135057, 7670326372532, 69286959112797, 647568753568636, 6251768635591613, 62255057942504968, 638658964709824185

Offset: 0

Author

Ilya Gutkovskiy, Feb 03 2022

Keywords

• nonn

Comments

Shifts 2 places left under 2nd-order binomial transform.

Crossrefs

Cf. A000995, A004211, A007472, A351053, A351128, A351132, A351143, A351161.

Programs

Maple

bintr:= proc(p) local b;
          b:= proc(n) option remember; add(p(k)*binomial(n, k), k=0..n) end
        end:
b:= (bintr@@2)(a):
a:= n-> `if`(n<2, n, b(n-2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 07 2025

Mathematica

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]
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}];
(* another pprogram *)
B[x_] := BesselK[0, 1]*BesselI[0, Exp[x]] - BesselI[0, 1]*BesselK[0, Exp[x]];
a[n_] := SeriesCoefficient[FullSimplify[Series[B[x], {x, 0, n}]], n] n!;
Table[a[n], {n, 0, 30}] (* Ven Popov, Apr 25 2025 *)

Formula

a(0) = 0, a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 2^k * a(n-k-2).

E.g.f.: BesselK(0, 1)*BesselI(0, exp(x)) - BesselI(0, 1)*BesselK(0, exp(x)). - Ven Popov, Apr 25 2025