A307678 G.f. A(x) satisfies: A(x) = 1 + x*A(x)^3/(1 - x).

1, 1, 4, 19, 101, 578, 3479, 21714, 139269, 912354, 6078832, 41066002, 280636657, 1936569717, 13475408847, 94446518559, 666149216744, 4724705621702, 33676421377532, 241100485812034, 1732999323835918, 12501487280292424, 90478497094713958, 656788523782034248, 4780725762185300389

Offset: 0

Author

Ilya Gutkovskiy, Apr 21 2019

Keywords

• nonn

Comments

Convolution square root of A270386.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 19*x^3 + 101*x^4 + 578*x^5 + 3479*x^6 + 21714*x^7 + 139269*x^8 + 912354*x^9 + 6078832*x^10 + ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1000
• Guillermo Esteban, Clemens Huemer, and Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.

Crossrefs

Cf. A001764, A002212, A006013, A127897, A188687 (partial sums), A270386.

Programs

Mathematica

terms = 24; A[] = 1; Do[A[x] = 1 + x A[x]^3/(1 - x) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[Sum[a[k] a[i - k] a[j - i], {k, 0, i}], {i, 0, j}], {j, 0, n - 1}]; Table[a[n], {n, 0, 24}]
terms = 24; CoefficientList[Series[2 Sqrt[(1 - x) Sin[1/3 ArcSin[3/2 Sqrt[3] Sqrt[x/(1 - x)]]]^2/x]/Sqrt[3], {x, 0, terms}], x]

Maxima

a(n):=sum(binomial(n-1,n-k)*(binomial(3*k,k))/(2*k+1),k,0,n); /* Vladimir Kruchinin, Feb 05 2022*/

PARI

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^3, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

Formula

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

a(n) ~ 31^(n + 1/2) / (3*sqrt(Pi) * n^(3/2) * 2^(2*n+2)). - Vaclav Kotesovec, May 06 2019

G.f.: (2/sqrt(3*x/(1-x)))*sin((1/3)*asin(sqrt((27*x/(1-x))/4))). - Vladimir Kruchinin, Feb 05 2022

a(n) = Sum_{k=0..n} C(n-1,n-k)*C(3*k,k)/(2*k+1). - Vladimir Kruchinin, Feb 05 2022