A349255 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - 2 * x * A(x)^2)).

1, 1, 7, 47, 369, 3113, 27631, 254239, 2403361, 23201393, 227771831, 2266983119, 22822484497, 231994748633, 2377894546783, 24548520253247, 255026759000897, 2664111200687969, 27967731861910759, 294900120348032623, 3121862973452544433, 33167268461833410569

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..950

Formula

a(n) = (-1)^n + 2 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).

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

a(n) = (-1)^n*hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], (3/2)^3). - Peter Luschny, Nov 12 2021

a(n) ~ sqrt(171 + 23*sqrt(57)) * (23 + 3*sqrt(57))^n / (9 * sqrt(Pi) * n^(3/2) * 2^(2*n + 5/2)). - Vaclav Kotesovec, Nov 13 2021

Mathematica

nmax = 21; A[_] = 0; Do[A[x_] = 1/((1 + x) (1 - 2 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = (-1)^n + 2 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n + k, n - k] 2^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 21}]
a[n_] := (-1)^n*HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, (3/2)^3]; Table[a[n], {n, 0, 21}] (* Peter Luschny, Nov 12 2021 *)

Crossrefs

Cf. A001003, A001764, A153231, A337168, A346627, A349253, A349254, A349256.

Sequence in context: A088057 A249477 A108434 * A093173 A368177 A218761

Adjacent sequences: A349252 A349253 A349254 * A349256 A349257 A349258

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 12 2021

Status

approved