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

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

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

Cf. A237646, A237647.

Sequence in context: A192778 A051138 A157101 * A091001 A297936 A298548

Adjacent sequences: A237645 A237646 A237647 * A237649 A237650 A237651

Keyword

nonn

Author

Paul D. Hanna, May 03 2014

Status

approved