A237648 - OEIS

%I #16 Mar 12 2015 22:34:54

%S 1,1,5,4,30,26,106,80,459,379,1451,1072,5210,4138,14894,10756,47617,

%T 36861,127949,91088,376264,285176,957336,672160,2640964,1968804,

%U 6452260,4483456,16921416,12437960,39873688,27435728,100259070,72823342,229410006,156586664,556880812,400294148

%N G.f. satisfies: A(x) = (1 + x + x^2) * A(x^2)^4.

%H Paul D. Hanna, <a href="/A237648/b237648.txt">Table of n, a(n) for n = 0..1000</a>

%F The 7th self-convolution yields A237647.

%F G.f. A(x) satisfies:

%F (1) A(x) = Product_{n>=0} ( 1 + x^(2^n) + x^(2*2^n) )^(4^n).

%F (2) A(x) / A(-x) = (1+x+x^2) / (1-x+x^2).

%F Bisections: let A(x) = B(x^2) + x*C(x^2), then

%F (3) B(x) = (1+x) * C(x).

%F (4) C(x) = A(x)^4 = (1+x+x^2)^4 * C(x^2)^4.

%e G.f.: A(x) = 1 + x + 5*x^2 + 4*x^3 + 30*x^4 + 26*x^5 + 106*x^6 + 80*x^7 + 459*x^8 +...

%e such that A(x) = (1+x+x^2) * A(x^2)^4, where:

%e A(x)^4 = 1 + 4*x + 26*x^2 + 80*x^3 + 379*x^4 + 1072*x^5 + 4138*x^6 + 10756*x^7 +...

%e The g.f. may thus be expressed by the product:

%e A(x) = (1+x+x^2) * (1+x^2+x^4)^4 * (1+x^4+x^8)^16 * (1+x^8+x^16)^64 *...

%e Note that x*A(x^2)^7 is the odd bisection of the g.f. G(x) of A237646:

%e A(x)^7 = 1 + 7*x + 56*x^2 + 273*x^3 + 1463*x^4 + 6048*x^5 + 26537*x^6 + 97903*x^7 +...+ A237647(n)*x^n +...

%e G(x) = (1+x+x^2)*A(x^2)^7 = 1 + x + 8*x^2 + 7*x^3 + 63*x^4 + 56*x^5 + 329*x^6 + 273*x^7 + 1736*x^8 + 1463*x^9 + 7511*x^10 + 6048*x^11 +...+ A237646(n)*x^n +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,#binary(n),A=(1+x+x^2)*subst(A^4,x,x^2) +x*O(x^n));polcoeff(A,n)}

%o for(n=0,50,print1(a(n),", "))

%o (PARI) {a(n)=local(A=1+x);A=prod(k=0,#binary(n),(1+x^(2^k)+x^(2*2^k)+x*O(x^n))^(4^k));polcoeff(A,n)}

%o for(n=0,50,print1(a(n),", "))

%Y Cf. A237646, A237647.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 03 2014