A351143 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 2*x)) / (1 - 2*x).

1, 0, 1, 2, 5, 16, 61, 258, 1177, 5776, 30537, 173394, 1050045, 6732608, 45459493, 322141106, 2390075249, 18525967328, 149684238801, 1257802518754, 10969260208565, 99100423076912, 926030783479629, 8937741026924450, 88988433270106249, 912906193294355952

Offset: 0

Author

Ilya Gutkovskiy, Feb 02 2022

Keywords

• nonn

Comments

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

Crossrefs

Cf. A000994, A004211, A007472, A351144, A351150, A351151, A351152.

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, 1-n, b(n-2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 07 2025

Mathematica

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]
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}]
(* another program *)
B[x_] := BesselK[1, 1]*BesselI[0, Exp[x]] + BesselI[1, 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) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 2^k * a(n-k-2).

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