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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.
4
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
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
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
FORMULA
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/Product_{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) = Product_{k>=1} (1-q^(n*k)). - Joerg Arndt, Mar 20 2010
Compare to the previous formula: 1/Product_{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 + ...
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)
MATHEMATICA
eta[q_]:= q^(1/24)*QPochhammer[q]; nmax = 250; a[n_]:=SeriesCoefficient[ Series[Exp[Sum[DivisorSigma[1, k]*2^(IntegerExponent[k, 2] + 1)*q^k/k, {k, 1, nmax}]], {q, 0, nmax}], n]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jul 03 2018 *)
nmax = 40; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*2^(IntegerExponent[k, 2] + 1)*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 20 2020 *)
nmax = 40; CoefficientList[Series[Product[1/EllipticTheta[4, 0, x^(2^k)]^(2^k), {k, 0, 1 + Log[2, nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 07 2023 *)
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. A163228 (B_0), A163229 (B_1), A029839 (T16B); variant: A163129. - Paul D. Hanna, Jul 26 2009
Sequence in context: A360323 A336127 A076508 * A100243 A026523 A066792
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 06 2009
STATUS
approved