OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1000
FORMULA
The 7th self-convolution yields A237647.
G.f. A(x) satisfies:
(1) A(x) = Product_{n>=0} ( 1 + x^(2^n) + x^(2*2^n) )^(4^n).
(2) A(x) / A(-x) = (1+x+x^2) / (1-x+x^2).
Bisections: let A(x) = B(x^2) + x*C(x^2), then
(3) B(x) = (1+x) * C(x).
(4) C(x) = A(x)^4 = (1+x+x^2)^4 * C(x^2)^4.
EXAMPLE
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 +...
such that A(x) = (1+x+x^2) * A(x^2)^4, where:
A(x)^4 = 1 + 4*x + 26*x^2 + 80*x^3 + 379*x^4 + 1072*x^5 + 4138*x^6 + 10756*x^7 +...
The g.f. may thus be expressed by the product:
A(x) = (1+x+x^2) * (1+x^2+x^4)^4 * (1+x^4+x^8)^16 * (1+x^8+x^16)^64 *...
Note that x*A(x^2)^7 is the odd bisection of the g.f. G(x) of A237646:
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 +...
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 +...
PROG
(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)}
for(n=0, 50, print1(a(n), ", "))
(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)}
for(n=0, 50, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 03 2014
STATUS
approved