A215623 - OEIS

%I #31 Sep 11 2024 05:47:37

%S 1,2,11,89,836,8551,92445,1039030,12019135,142151324,1711116646,

%T 20894534324,258195565959,3222677162409,40569811695707,

%U 514520507077695,6567611974106756,84310605465652750,1087798325715407703,14098475168420865396,183465816241394787196

%N G.f. satisfies A(x) = (1 + x*A(x)) * (1 + x*A(x)^4).

%C The radius of convergence of g.f. A(x) is r = 0.0712256396327314729661274986100... with A(r) = 1.4248895273944523042559975726479124492235978714420... where y=A(r) satisfies 3*y^7 - 4*y^6 + 16*y^5 - 28*y^4 + 8*y^3 - 4 = 0.

%H Seiichi Manyama, <a href="/A215623/b215623.txt">Table of n, a(n) for n = 0..876</a>

%F G.f. satisfies A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * A(x)^(3*k)).

%F The formal inverse of g.f. A(x) is (sqrt((1-x^3)^2 + 4*x^4) - (1+x^3))/(2*x^4).

%F a(n) = Sum_{k=0..n} binomial(n+3*k+1,k) * binomial(n+3*k+1,n-k) / (n+3*k+1). - _Seiichi Manyama_, Jul 19 2023

%F From  _Peter Bala_, Sep 10 2024: (Start)

%F x/series_reversion(x*A(x)) = 1 + 2*x + 7*x^2 + 39*x^3 + 242*x^4 + 1634*x^5 + ..., the g.f. of A364336.

%F (1/x) * series_reversion(x/A(x)) = 1 + 2*x + 15*x^2 + 163*x^3 + 2070*x^4 + 28698*x^5 + ..., the g.f. of A364331. (End)

%e G.f.: A(x) = 1 + 2*x + 11*x^2 + 89*x^3 + 836*x^4 + 8551*x^5 + 92445*x^6 + ...

%e Related expansions.

%e A(x)^4 = 1 + 8*x + 68*x^2 + 652*x^3 + 6750*x^4 + 73544*x^5 + 831078*x^6 + ...

%e A(x)^5 = 1 + 10*x + 95*x^2 + 965*x^3 + 10350*x^4 + 115507*x^5 + ...

%e where A(x) = 1 + x*(A(x) + A(x)^4) + x^2*A(x)^5.

%e The logarithm of the g.f. equals the series:

%e log(A(x)) = (1 + A(x)^3)*x + (1 + 2^2*A(x)^3 + A(x)^6)*x^2/2 +

%e   (1 + 3^2*A(x)^3 + 3^2*A(x)^6 + A(x)^9)*x^3/3 +

%e   (1 + 4^2*A(x)^3 + 6^2*A(x)^6 + 4^2*A(x)^9 + A(x)^12)*x^4/4 +

%e   (1 + 5^2*A(x)^3 + 10^2*A(x)^6 + 10^2*A(x)^9 + 5^2*A(x)^12 + A(x)^15)*x^5/5 + ...

%e more explicitly,

%e log(A(x)) = 2*x + 18*x^2/2 + 209*x^3/3 + 2550*x^4/4 + 32082*x^5/5 + 411705*x^6/6 + 5356416*x^7/7 + ....

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(A+x*O(x^n))^(3*j))*x^m/m))); polcoeff(A, n)}

%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1+x*A)*(1+x*A^4)+x*O(x^n)); polcoeff(A, n)}

%o for(n=0,21,print1(a(n),", "))

%Y Cf. A198953, A215624, A036765, A198951, A181734, A364331, A364336, A364376.

%K nonn,easy

%O 0,2

%A _Paul D. Hanna_, Aug 17 2012