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A237648 G.f. satisfies: A(x) = (1 + x + x^2) * A(x^2)^4.

%I A237648 #16 Mar 12 2015 22:34:54
%S A237648 1,1,5,4,30,26,106,80,459,379,1451,1072,5210,4138,14894,10756,47617,
%T A237648 36861,127949,91088,376264,285176,957336,672160,2640964,1968804,
%U A237648 6452260,4483456,16921416,12437960,39873688,27435728,100259070,72823342,229410006,156586664,556880812,400294148
%N A237648 G.f. satisfies: A(x) = (1 + x + x^2) * A(x^2)^4.
%H A237648 Paul D. Hanna, <a href="/A237648/b237648.txt">Table of n, a(n) for n = 0..1000</a>
%F A237648 The 7th self-convolution yields A237647.
%F A237648 G.f. A(x) satisfies:
%F A237648 (1) A(x) = Product_{n>=0} ( 1 + x^(2^n) + x^(2*2^n) )^(4^n).
%F A237648 (2) A(x) / A(-x) = (1+x+x^2) / (1-x+x^2).
%F A237648 Bisections: let A(x) = B(x^2) + x*C(x^2), then
%F A237648 (3) B(x) = (1+x) * C(x).
%F A237648 (4) C(x) = A(x)^4 = (1+x+x^2)^4 * C(x^2)^4.
%e A237648 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 A237648 such that A(x) = (1+x+x^2) * A(x^2)^4, where:
%e A237648 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 A237648 The g.f. may thus be expressed by the product:
%e A237648 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 A237648 Note that x*A(x^2)^7 is the odd bisection of the g.f. G(x) of A237646:
%e A237648 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 A237648 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 A237648 (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 A237648 for(n=0,50,print1(a(n),", "))
%o A237648 (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 A237648 for(n=0,50,print1(a(n),", "))
%Y A237648 Cf. A237646, A237647.
%K A237648 nonn
%O A237648 0,3
%A A237648 _Paul D. Hanna_, May 03 2014