A351053 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 3*x)) / (1 - 3*x).

0, 1, 0, 1, 6, 28, 126, 613, 3438, 22159, 157362, 1189126, 9436320, 78690781, 692478684, 6439539457, 63106488618, 648453907216, 6952719052134, 77521908188737, 897132401326458, 10764085132255807, 133774484448519294, 1720018195807299418, 22847325911461934352

Offset: 0

Author

Ilya Gutkovskiy, Feb 03 2022

Keywords

• nonn

Comments

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

Crossrefs

Cf. A000995, A004212, A351028, A351049, A351128, A351132, A351144, A351161.

Programs

Mathematica

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

Formula

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