A353042 G.f. A(x) satisfies: A(x) = 1 + x * A(4*x/(1 + 3*x)) / (1 - x).

1, 1, 5, 73, 4301, 1065361, 1079026325, 4404504773593, 72088402948928861, 4722943066827454121761, 1237982543178169058402322725, 1298086594246614900499652230482793, 5444532149619463867564918804810528611821, 91343917667481554378430257939829428893551284401

Offset: 0

Author

Ilya Gutkovskiy, Apr 19 2022

Keywords

• nonn

Crossrefs

Cf. A006898, A053763, A135756 (partial sums), A353041.

Programs

Mathematica

nmax = 13; A[] = 0; Do[A[x] = 1 + x A[4 x/(1 + 3 x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] 2^(k (k - 1)), {k, 0, n}], {n, 0, 13}]

Formula

G.f.: Sum_{k>=0} 2^(k*(k-1)) * (x/(1 - x))^k.

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