A248876 - OEIS

%I #17 Mar 08 2015 21:00:50

%S 1,1,1,2,4,8,13,24,45,85,161,305,582,1116,2149,4152,8049,15653,30528,

%T 59695,117012,229880,452565,892703,1764099,3492029,6923494,13747483,

%U 27335873,54427621,108505081,216568556,432740907,865610375,1733227339,3473805680,6968708734,13991916510,28116598325

%N G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..3*n} T(n,k)^2 * x^k] / A(x)^n * x^n/n ) where T(n,k) equals the coefficient of x^k in (1+x+x^2+x^3)^n.

%C Compare the definition of this sequence to G(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / G(x)^n * x^n/n ) where T(n,k) = [x^k] (1+x+x^2)^n, which is satisfied by the rational function: G(x) = (1+x^3)*(1+x^6)/((1-x)*(1-x^4)).

%H Paul D. Hanna, <a href="/A248876/b248876.txt">Table of n, a(n) for n = 0..100</a>

%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 13*x^6 + 24*x^7 +...

%e where

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

%e (1 + 2^2*x + 3^2*x^2 + 4^2*x^3 + 3^2*x^4 + 2^2*x^5 + x^6)/A(x)^2 * x^2/2 +

%e (1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 12^2*x^4 + 12^2*x^5 + 10^2*x^6 + 6^2*x^7 + 3^2*x^8 + x^9)/A(x)^3 * x^3/3 +

%e (1 + 4^2*x + 10^2*x^2 + 20^2*x^3 + 31^2*x^4 + 40^2*x^5 + 44^2*x^6 + 40^2*x^7 + 31^2*x^8 + 20^2*x^9 + 10^2*x^10 + 4^2*x^11 + x^12)/A(x)^4 * x^4/4 +

%e which involves the squares of the coefficients in (1 + x + x^2 + x^3)^n.

%o (PARI) /* By Definition: */

%o {T(n,k)=polcoeff((1 + x + x^2 + x^3 + x*O(x^k))^n, k)}

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

%o for(n=0, 40, print1(a(n), ", "))

%Y Cf. A255839, A200537, A251687, A251688, A005721.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Mar 04 2015