A348793 G.f. A(x) satisfies A(x) = (1 + x * A(x)^3) / (1 - 2 * x).

1, 3, 15, 102, 807, 6951, 63240, 597864, 5815167, 57815553, 584919951, 6002197914, 62321630100, 653553174756, 6912106219176, 73642451396160, 789642274208271, 8515008918555573, 92281921130853213, 1004600177464845450, 10980406558088695599, 120454756647900759543

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..937

Formula

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

a(n) ~ sqrt((2 + s^3)/(3*Pi*s*(1 - 2*r))) / (2*n^(3/2)*r^n), where r = (2 + (3*(-2 + sqrt(6))^(1/3))/2^(2/3) - 3/(2*(-2 + sqrt(6)))^(1/3)) / 4 = 0.084819663336750180604484695162155813902734598764355... and s = 1/2 + (-sqrt(2) + sqrt(3))/(2^(5/6)*(-2 + sqrt(6))^(1/3)) + 1/(2*(-2 + sqrt(6)))^(2/3) = 1.8064439323587723772036249693148814564378856424032... - Vaclav Kotesovec, Nov 04 2021

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(n+2*k+1,n) / (n+2*k+1). - Seiichi Manyama, Jul 24 2023

Mathematica

nmax = 21; A[_] = 0; Do[A[x_] = (1 + x A[x]^3)/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + 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}]

Prog

(PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(n+2*k+1, n)/(n+2*k+1)); \\ Seiichi Manyama, Jul 24 2023

Crossrefs

Cf. A001764, A047891, A346626.

Sequence in context: A109777 A242003 A265164 * A135903 A185753 A174493

Adjacent sequences: A348790 A348791 A348792 * A348794 A348795 A348796

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 03 2021

Status

approved