A349256 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - 3 * x * A(x)^2)).

1, 2, 19, 206, 2563, 34415, 486370, 7128488, 107364421, 1651615568, 25840137724, 409898503763, 6577319627506, 106571487893024, 1741193467526782, 28653852176675324, 474521786894159593, 7902112425718228064, 132243695376774536755, 2222925664652778182060

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..797

Formula

a(n) = (-1)^n + 3 * 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) * 3^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)^4). - Peter Luschny, Nov 12 2021

a(n) ~ sqrt(585 + 73*sqrt(65)) * (73 + 9*sqrt(65))^n / (3^(5/2) * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - Vaclav Kotesovec, Nov 13 2021

Mathematica

nmax = 19; A[_] = 0; Do[A[x_] = 1/((1 + x) (1 - 3 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = (-1)^n + 3 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, 19}]
Table[Sum[(-1)^(n - k) Binomial[n + k, n - k] 3^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 19}]
a[n_] := (-1)^n*HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, (3/2)^4]; Table[a[n], {n, 0, 19}] (* Peter Luschny, Nov 12 2021 *)

Crossrefs

Cf. A001764, A107841, A217363, A337169, A346627, A349253, A349254, A349255.

Sequence in context: A268707 A037071 A126039 * A091852 A210986 A232525

Adjacent sequences: A349253 A349254 A349255 * A349257 A349258 A349259

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 12 2021

Status

approved