A351144 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 3*x)) / (1 - 3*x).

1, 0, 1, 3, 10, 39, 181, 972, 5797, 37389, 258202, 1905681, 15016465, 125920872, 1117950913, 10452866439, 102485649754, 1050464300187, 11231883627301, 125055844922916, 1447371528438565, 17382103226123313, 216221862096537994, 2781342531957176085, 36942930754308211969

Offset: 0

Author

Ilya Gutkovskiy, Feb 02 2022

Keywords

• nonn

Comments

Shifts 2 places left under 3rd-order binomial transform.

Crossrefs

Cf. A000994, A004212, A351049, A351143, A351150, A351151, A351152.

Programs

Mathematica

nmax = 24; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 3 x)]/(1 - 3 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] 3^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 24}]

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 3^k * a(n-k-2).