A304936 a(n) = [x^n] 1/(1 - n*x/(1 - x - n*x/(1 - x - n*x/(1 - x - n*x/(1 - x - n*x/(1 - ...)))))), a continued fraction.

1, 1, 10, 183, 5076, 191105, 9140118, 531731935, 36496595656, 2889768574449, 259443165181410, 26054614893427703, 2894791106297891100, 352618782117325104849, 46736101530152250554926, 6696645353339606889836415, 1031600569146491935984293648, 170029083604373881344301895585

Offset: 0

Author

Ilya Gutkovskiy, May 21 2018

Keywords

• nonn

Crossrefs

Cf. A001003, A107841, A131763, A131765, A131846, A131869, A131926, A131927, A292798.

Programs

Mathematica

Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-n x, 1 - x, {i, 1, n}]), {x, 0, n}], {n, 0, 17}]
Table[SeriesCoefficient[2/(1 + x + Sqrt[1 - x (2 + 4 n - x)]), {x, 0, n}], {n, 0, 17}]
Table[Sum[(-1)^(n - k) (n + 1)^k Binomial[n, k] Binomial[n + k, k]/(k + 1),{k, 0, n}], {n, 0, 17}]
Table[(-1)^n Hypergeometric2F1[-n, n + 1, 2, n + 1], {n, 0, 17}]

Formula

a(n) = [x^n] 2/(1 + x + sqrt(1 - x*(2 + 4*n - x))).

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

a(n) ~ exp(1/2) * 2^(2*n) * n^(n - 3/2) / sqrt(Pi). - Vaclav Kotesovec, Jun 08 2019