A274379 - OEIS

%I #20 Jan 28 2024 07:26:28

%S 1,3,13,70,429,2842,19794,142758,1056655,7980280,61251261,476387379,

%T 3746317414,29738316330,237968639936,1917578268288,15546796822656,

%U 126728260011920,1037987924978125,8538459191677170,70509828893263474,584310452973463242,4857624566855734836,40501472981905806550,338594135314564168494,2837641019938074131463,23835438376045780734390,200633658871150345742269,1692132786239339256115050,14297391426538004065333910,121009206594941545408186768

%N G.f. satisfies A(x) = (1 + x*A(x))^3 * (1 + x^2*A(x)^3).

%H Seiichi Manyama, <a href="/A274379/b274379.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. satisfies: A(x) = (1/x) * Series_Reversion( x*(1 - x^2*(1+x)^3) / (1+x)^3 ).

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

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(3*n+3*k+3,n-2*k). - _Seiichi Manyama_, Jan 27 2024

%e  G.f.: A(x) = 1 + 3*x + 13*x^2 + 70*x^3 + 429*x^4 + 2842*x^5 + 19794*x^6 + 142758*x^7 + 1056655*x^8 + 7980280*x^9 + ...

%e such that A(x) = 1 + 3*x*A(x) + x^2*(3*A(x)^2 + A(x)^3) + x^3*(A(x)^3 + 3*A(x)^4) + 3*x^4*A(x)^5 + x^5*A(x)^6.

%o (PARI) {a(n) = my(A=1); for(i=1, n, A = (1 + x*A)^3 * (1 + x^2*A^3) + x*O(x^n) ); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n) = my(A=1); A = (1/x)*serreverse(x*(1-x^2*(1+x)^3)/(1+x +x^2*O(x^n) )^3 ); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A181734, A274378.

%Y Cf. A274735, A369600.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 04 2016