A307678 - OEIS

%I #26 Dec 09 2024 04:48:15

%S 1,1,4,19,101,578,3479,21714,139269,912354,6078832,41066002,280636657,

%T 1936569717,13475408847,94446518559,666149216744,4724705621702,

%U 33676421377532,241100485812034,1732999323835918,12501487280292424,90478497094713958,656788523782034248,4780725762185300389

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

%C Convolution square root of A270386.

%H Michael De Vlieger, <a href="/A307678/b307678.txt">Table of n, a(n) for n = 0..1000</a>

%H Guillermo Esteban, Clemens Huemer, and Rodrigo I. Silveira, <a href="https://arxiv.org/abs/2003.00524">New production matrices for geometric graphs</a>, arXiv:2003.00524 [math.CO], 2020.

%F 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).

%F a(n) ~ 31^(n + 1/2) / (3*sqrt(Pi) * n^(3/2) * 2^(2*n+2)). - _Vaclav Kotesovec_, May 06 2019

%F G.f.: (2/sqrt(3*x/(1-x)))*sin((1/3)*asin(sqrt((27*x/(1-x))/4))). - _Vladimir Kruchinin_, Feb 05 2022

%F a(n) = Sum_{k=0..n} C(n-1,n-k)*C(3*k,k)/(2*k+1). - _Vladimir Kruchinin_, Feb 05 2022

%e 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 + ...

%t 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]

%t 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}]

%t 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]

%o (Maxima)

%o a(n):=sum(binomial(n-1,n-k)*(binomial(3*k,k))/(2*k+1),k,0,n); /* _Vladimir Kruchinin_, Feb 05 2022*/

%o (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);

%o A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^3, k)) )); A[n+1]}

%o for(n=0, 30, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Nov 23 2024, after _Paul D. Hanna_

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 21 2019