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A237648
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G.f. satisfies: A(x) = (1 + x + x^2) * A(x^2)^4.
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3
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1, 1, 5, 4, 30, 26, 106, 80, 459, 379, 1451, 1072, 5210, 4138, 14894, 10756, 47617, 36861, 127949, 91088, 376264, 285176, 957336, 672160, 2640964, 1968804, 6452260, 4483456, 16921416, 12437960, 39873688, 27435728, 100259070, 72823342, 229410006, 156586664, 556880812, 400294148
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OFFSET
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0,3
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LINKS
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FORMULA
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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.
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EXAMPLE
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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 +...
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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