A167002 G.f.: A(x) = Sum_{n>=0} 2^n*log(G(2^n*x))^n/n! where G(x) = g.f. of A167000.

1, 4, 20, 320, 21064, 5030400, 4056470528, 10872157339648, 98162974155542592, 3052890463194814939136, 334052589949087491382968320, 130858881562759880830581892710400

Offset

0,2

Comments

The g.f. of A167000, G(x), satisfies:

Sum_{n>=0} log(G(2^n*x))^n/n! = 1 + Sum_{n>=0} 2^(2^n-n)*x^(2^n).

Links

Table of n, a(n) for n=0..11.

Formula

a(n) = [x^n] G(x)^(2^(n+1)) for n>=0 where G(x) = g.f. of A167000.

Example

G.f.: A(x) = 1 + 4*x + 20*x^2 + 320*x^3 + 21064*x^4 + 5030400*x^5 +...
Let G(x) equal the g.f. of A167000:
G(x) = 1 + x - x^2 - 16*x^4 - 1767*x^5 - 493164*x^6 - 422963721*x^7 +...
then the g.f. A(x) of this sequence equals the series:
A(x) = 1 + 2*log(G(2x)) + 4*log(G(4x))^2/2! + 8*log(G(8x))^3/3! + 16*log(G(16x))^4/4! +...
ILLUSTRATE (2^n)-th POWERS OF G.F. G(x) OF A167000.
The coefficients in the expansion of G(x)^(2^n), n>=0, begin:
G^1: [1,1,-1,0,-16,-1767,-493164,-422963721,-1130568823448,...];
G^2: [(1),2,-1,-2,-31,-3566,-989830,-846910236,...];
G^4: [1,(4),2,-8,-69,-7252,-1993858,-1697772536,...];
G^8: [1,8,(20),0,-198,-15088,-4045944,-3411523840,...];
G^16: [1,16,104,(320),4,-33344,-8341216,-6888386304,...];
G^32: [1,32,464,3968,(21064),0,-17646208,-14050624512,...];
G^64: [1,64,1952,37632,511376,(5030400),0,-29063442432,...];
G^128: [1,128,8000,325120,9649952,222432256,(4056470528),0,...]; ...
where the coefficients along the diagonal (shown in parenthesis) form the initial terms of this sequence.

Prog

(PARI) {a(n)=local(A=[1, 4], B=[1, 2], G=[1, 1]); for(i=1, n, G=concat(G, 0); B=Vec(sum(m=0, #G, log(subst(Ser(G), x, 2^m*x))^m/m!)); G[ #G]=-floor(B[ #G]/2^(#G-1))); A=Vec(sum(m=0, #G, 2^m*log(subst(Ser(G), x, 2^m*x))^m/m!)); A[n+1]}

Crossrefs

Cf. A167000, A167001.

Sequence in context: A012797 A342907 A358544 * A227005 A054465 A118713

Adjacent sequences: A166999 A167000 A167001 * A167003 A167004 A167005

Keyword

nonn

Author

Paul D. Hanna, Nov 14 2009

Status

approved