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A162584 G.f.: A(x) = exp( 2*Sum_{n>=1} sigma(n)*A006519(n) * x^n/n ), where A006519(n) = highest power of 2 dividing n. 3
1, 2, 8, 16, 50, 96, 240, 448, 1024, 1858, 3888, 6896, 13696, 23776, 44960, 76608, 139970, 234432, 414904, 684336, 1181568, 1921472, 3242928, 5206208, 8623104, 13679490, 22268752, 34941120, 56039936, 87036576, 137686048, 211822976 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Log of the g.f. A(x) is formed from the term-wise product of the log of the g.f.s of the partition numbers A000041 and the binary partitions A000123.

LINKS

Table of n, a(n) for n=0..31.

FORMULA

Contribution from Paul D. Hanna, Jul 26 2009: (Start)

Define series BISECTIONS A(q) = B_0(q) + B_1(q), then

2*B_0(q)/B_1(q) = T16B(q) = q*eta(q^8)^6/(eta(q^4)^2*eta(q^16)^4),

the McKay-Thompson series of class 16B for the Monster group (A029839). (End)

G.f.: 1/prod(n>=0, Theta4(q^(2^n))^(2^n) ) = 1 / ( E(1)^2*E(2)^3*E(4)^6*E(8)^12* ... * E(2^n)^A042950(n) * ... ) where E(n) = prod(k>=1, 1-q^(n*k) ). [Joerg Arndt, Mar 20 2010]

Compare to the previous formula: 1/prod(n>=0, Theta3(q^(2^n))^(2^n) ) = Theta4(q). [Joerg Arndt, Aug 03 2011]

EXAMPLE

G.f.: A(x) = 1 + 2*x + 8*x^2 + 16*x^3 + 50*x^4 + 96*x^5 + 240*x^6 +...

log(A(x))/2 = x + 6*x^2/2 + 4*x^3/3 + 28*x^4/4 + 6*x^5/5 + 24*x^6/6 + 8*x^7/7 + 120*x^8/8 +...+ sigma(n)*A006519(n)*x^n/n +...

The log of the g.f. of the Partition numbers (A000041) is:

x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 6*x^5/5 + 12*x^6/6 +...+ sigma(n)*x^n/n +...

The log of the g.f. of the binary partitions (A000123) is:

x + x^2/2 + x^3/3 + 4*x^4/4 + x^5/5 + 2*x^6/6 + x^7/7 +...+ A006519(n)*x^n/n +...

Contribution from Paul D. Hanna, Jul 26 2009: (Start)

BISECTIONS begin:

B_0(q) = 1 + 8*q^2 + 50*q^4 + 240*q^6 + 1024*q^8 + 3888*q^10 +...

B_1(q) = 2*q + 16*q^3 + 96*q^5 + 448*q^7 + 1858*q^9 + 6896*q^11 +...

where 2*B_0(q)/B_1(q) = T16B(q):

T16B = 1/q + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 +...

which is a g.f. of A029839. (End)

PROG

(PARI) {a(n)=local(L=sum(m=1, n, 2*sigma(m)*2^valuation(m, 2)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}

CROSSREFS

Cf. A000203, A006519, A000041, A000123.

Cf. A163228 (B_0), A163229 (B_1), A029839 (T16B); variant: A163129. [From Paul D. Hanna, Jul 26 2009]

Sequence in context: A134353 A280229 A076508 * A100243 A026523 A066792

Adjacent sequences:  A162581 A162582 A162583 * A162585 A162586 A162587

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 06 2009

STATUS

approved

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Last modified September 26 04:27 EDT 2017. Contains 292502 sequences.