A155200 -id:A155200 - cp's OEIS frontend

A166168 G.f.: exp( Sum_{n>=1} Lucas(n^2)*x^n/n ) where Lucas(n) = A000204(n).

1, 1, 4, 29, 585, 34212, 5600397, 2490542953, 2968152042068, 9416588994339205, 79216509536543420965, 1762508872870620792746360, 103525263562786817866762466405, 16031370626878431551103688398524485

Offset: 0

Paul D. Hanna, Oct 08 2009

nonn

Conjectured to consist entirely of integers.
The Lucas numbers (A000204) forms the logarithmic derivative of the Fibonacci numbers (A000045).
Note that Lucas(n^2) = [(1+sqrt(5))/2]^(n^2) + [(1-sqrt(5))/2]^(n^2).

			G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 585*x^4 + 34212*x^5 +...
log(A(x)) = x + 7*x^2/2 + 76*x^3/3 + 2207*x^4/4 + 167761*x^5/5 + 33385282*x^6/6 +...+ Lucas(n^2)*x^n/n +...

G. C. Greubel, Table of n, a(n) for n = 0..60
Sawian Jaidee, Patrick Moss, Tom Ward, Time-changes preserving zeta functions, arXiv:1809.09199 [math.DS], 2018.

Cf. A166169, A156216, A155200, A000204, A000045.

with(combinat): seq(coeff(series(exp(add((fibonacci(k^2-1)+fibonacci(k^2+1))*x^k/k,k=1..n)),x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Dec 18 2018

CoefficientList[Series[Exp[Sum[LucasL[n^2]*x^n/n, {n, 1, 200}]], {x, 0, 50}], x](* G. C. Greubel, May 06 2016 *)

{a(n)=polcoeff(exp(sum(m=1,n,(fibonacci(m^2-1)+fibonacci(m^2+1))*x^m/m)+x*O(x^n)),n)}

a(n) = (1/n)*Sum_{k=1..n} Lucas(k^2)*a(n-k), a(0)=1.

Logarithmic derivative yields A166169.

A156171 G.f.: A(x) = exp( Sum_{n>=1} x^n/(1 - 2^n*x)^n / n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 3, 11, 53, 357, 3521, 51665, 1122135, 35638903, 1639453459, 108526044099, 10298220348807, 1396920580458279, 270394562069007327, 74574294532698008703, 29276455806256470979269, 16344863466384180848085765, 12969208162308705691408055345, 14616452655308018025267503353697

Offset: 0

Paul D. Hanna, Feb 05 2009

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 53*x^4 + 357*x^5 + 3521*x^6 + 51665*x^7 + 1122135*x^8 + 35638903*x^9 + 1639453459*x^10 + 108526044099*x^11 +...
such that:
log(A(x)) = Sum_{n>=1} x^n/n * (1 + 2^n*x + 4^n*x^2 +...+ 2^(n*k)*x^k +...)^n
or
log(A(x)) = x*(1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 32*x^5 +...) +
x^2/2*(1 + 8*x + 48*x^2 + 256*x^3 + 1280*x^4 + 6144*x^5 +...) +
x^3/3*(1 + 24*x + 384*x^2 + 5120*x^3 + 61440*x^4 + 688128*x^5 +...) +
x^4/4*(1 + 64*x + 2560*x^2 + 81920*x^3 + 2293760*x^4 + 58720256*x^5 +...) +
x^5/5*(1 + 160*x + 15360*x^2 + 1146880*x^3 + 73400320*x^4 + 4227858432*x^5 +...) +
x^6/6*(1 + 384*x + 86016*x^2 + 14680064*x^3 + 2113929216*x^4 + 270582939648*x^5 +...) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 161*x^4/4 + 1441*x^5/5 + 18305*x^6/6 + 330625*x^7/7 + 8488961*x^8/8 + 309465601*x^9/9 + 16011372545*x^10/10 + 1174870185985*x^11/11 + 122233833963521*x^12/12 +...

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A156170, A155200, A156100.

nmax = 20; CoefficientList[Series[Exp[Sum[x^k/(1 - 2^k*x)^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 17 2020 *)

{a(n)=polcoeff(exp(sum(m=1,n,x^m/(1-2^m*x+x*O(x^n))^m/m)),n)}

a(n) ~ c * 2^(n^2/4 + n + 1/2) / (sqrt(Pi) * n^(3/2)), where c = EllipticTheta[3, 0, 1/2] = JacobiTheta3(0,1/2) = 2.1289368272118771586694585... if n is even and c = EllipticTheta[2, 0, 1/2] = JacobiTheta2(0,1/2) = 2.1289312505130275585916134... if n is odd. - Vaclav Kotesovec, Oct 17 2020

A165937 G.f.: A(x) = exp( Sum_{n>=1} A002203(n^2)*x^n/n ).

Original entry on oeis.org

1, 2, 19, 964, 334965, 742714950, 10042408885191, 814556580116590856, 393147641272746246076745, 1123539400297807898234860367690, 18948227277012085227250633551784337179, 1881331163508674280605070386666674939623268684

Offset: 0

Paul D. Hanna, Oct 18 2009

nonn

A002203 equals the logarithmic derivative of the Pell numbers (A000129).
Note that A002203(n^2) = (1+sqrt(2))^(n^2) + (1-sqrt(2))^(n^2).
Given g.f. A(x), (1-x)^(1/4) * A(x)^(1/8) is an integer series.

			G.f.: A(x) = 1 + 2*x + 19*x^2 + 964*x^3 + 334965*x^4 + 742714950*x^5 +...
log(A(x)) = 2*x + 34*x^2/2 + 2786*x^3/3 + 1331714*x^4/4 + 3710155682*x^5/5 + 60245508192802*x^6/6 + 5701755387019728962*x^7/7 +...+ A002203(n^2)*x^n/n +...

Cf. A165938, A166879, A002203, A000129; variants: A158843, A166168, A155200.

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^(m^2))),m^2)*x^m/m)+x*O(x^(n^2))),n))}

Logarithmic derivative equals A165938.

Self-convolution of A166879.

A202518 G.f. satisfies: A(x) = exp( Sum_{n>=1} (2^n - A(x))^n * x^n/n ).

Original entry on oeis.org

1, 1, 4, 111, 12600, 5722258, 10419647136, 76124127132667, 2234758718926030048, 263964471372716219981614, 125532541357451846737479404864, 240382906462440786858510574342553910, 1852958218856132372722626702327036659515008

Offset: 0

Paul D. Hanna, Dec 20 2011

nonn

Compare g.f. with: G(x) = exp(Sum_{n>=1} (2 - G(x))^n * x^n/n) = 1 + x*C(-x^2) where C(x) is the Catalan function (A000108).

			G.f.: A(x) = 1 + x + 4*x^2 + 111*x^3 + 12600*x^4 + 5722258*x^5 +...
where
log(A(x)) = (2 - A(x))*x + (2^2 - A(x))^2*x^2/2 + (2^3 - A(x))^3*x^3/3 + (2^4 - A(x))^4*x^4/4 +...

Cf. A163138, A155200.

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,(2^m-A+x*O(x^n))^m*x^m/m)));polcoeff(A,n)}

A211892 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n^2) * x^n/n ), where Jacobsthal(n) = A001045(n).

Original entry on oeis.org

1, 3, 12, 198, 16962, 6762210, 11473594848, 80455865485692, 2306084412391039038, 268657100633050977422322, 126765866001055606588876061400, 241678197713843578271875740922972788, 1858396158245858742065123341776166504084452

Offset: 0

Paul D. Hanna, Apr 24 2012

nonn

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

			G.f.: A(x) = 1 + 3*x + 12*x^2 + 198*x^3 + 16962*x^4 + 6762210*x^5 +...
such that
log(A(x))/3 = x + 5*x^2/2 + 171*x^3/3 + 21845*x^4/4 + 11184811*x^5/5 + 22906492245*x^6/6 + 187649984473771*x^7/7 +...+ Jacobsthal(n^2)*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,...].

Cf. A211893, A211894, A211895, A211896, A207972, A001045, A155200.

Cf. A231279 (Jacobsthal(n^2)).

{Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k^2)*x^k/k)+x*O(x^n)), n)}
for(n=0, 16, print1(a(n), ", "))

G.f.: (1+x) * exp( Sum_{n>=1} 2^(n^2) * x^n/n ).

a(n) = A155200(n) + A155200(n-1).

A292500 G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} (2k-1)^n x^k ]^n / n ).

Original entry on oeis.org

1, 1, 4, 18, 122, 1382, 26992, 967860, 59207134, 6539607238, 1225903048760, 407719392472476, 233686070341415140, 233030334505100451484, 407716349332865096406960, 1219594666823043463552070760, 6484753389847998264537623184230, 58288150472645787928029816422705798, 936721167715228772497787011017302901192, 25340260842241991639562678352357479545874188

Offset: 0

Paul D. Hanna, Sep 17 2017

nonn

A060187(n,k) = Sum_{j=1..k} (-1)^(k-j) * binomial(n,k-j) * (2*j-1)^(n-1).
Note that exp( Sum_{n>=1} [ Sum_{k=0..n} A060187(n+1,k+1) * x^k ]  / (1-x)^(n+1) * x^n/n ) does not yield an integer series.
Conjecture: a(n)^(1/n^2) tends to 3^(1/4). - Vaclav Kotesovec, Oct 17 2020

			G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 122*x^4 + 1382*x^5 + 26992*x^6 + 967860*x^7 + 59207134*x^8 + 6539607238*x^9 + 1225903048760*x^10 + 407719392472476*x^11 + 233686070341415140*x^12 + 233030334505100451484*x^13 + 407716349332865096406960*x^14 + 1219594666823043463552070760*x^15 +...
RELATED SERIES.
log(A(x)) = x + 7*x^2/2 + 43*x^3/3 + 399*x^4/4 + 6091*x^5/5 + 151255*x^6/6 + 6550307*x^7/7 + 465127199*x^8/8 + 58293976795*x^9/9 + 12191724780647*x^10/10 + 4471204259257363*x^11/11 + 2799295142330495151*x^12/12 + 3026340345288168023883*x^13/13 + 5704756586858875194533367*x^14/14 +...+ A292502(n)*x^n/n +...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 3^n*x^2 + 5^n*x^3 +...+ (2*k-1)^n*x^k +...)^n/n,
or,
log(A(x)) = (x + 3*x^2 + 5*x^3 + 7*x^4 + 9*x^5 +...) +
(x + 3^2*x^2 + 5^2*x^3 + 7^2*x^4 + 9^2*x^5 +...)^2/2 +
(x + 3^3*x^2 + 5^3*x^3 + 7^3*x^4 + 9^3*x^5 +...)^3/3 +
(x + 3^4*x^2 + 5^4*x^3 + 7^4*x^4 + 9^4*x^5 +...)^4/4 + ...
This logarithmic series can be written using the Eulerian numbers of type B like so:
log(A(x)) = (x + x^2) / (1-x)^2 +
(x + 6*x^2 + x^3)^2 / (1-x)^6/2 +
(x + 23*x^2 + 23*x^3 + x^4)^3 / (1-x)^12/3 +
(x + 76*x^2 + 230*x^3 + 76*x^4 + x^5)^4 / (1-x)^20/4 +
(x + 237*x^2 + 1682*x^3 + 1682*x^4 + 237*x^5 + x^6)^5 / (1-x)^30/5 +
(x + 722*x^2 + 10543*x^3 + 23548*x^4 + 10543*x^5 + 722*x^6 + x^7)^6 / (1-x)^42/6 +
(x + 2179*x^2 + 60657*x^3 + 259723*x^4 + 259723*x^5 + 60657*x^6 + 2179*x^7 + x^8)^7 / (1-x)^56/7 +...+
[ Sum_{k=0..n} A060187(n+1,k+1) * x^k ]^n / (1-x)^(n^2+n) * x^n/n +...

Paul D. Hanna, Table of n, a(n) for n = 0..130

Cf. A292501, A292502 (log), A156170, A276751, A276752, A156171, A155200, A060187.

nmax = 20; CoefficientList[Series[Exp[Sum[2^(k^2) * x^k * LerchPhi[x, -k, 1/2]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 17 2020 *)

{a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n+1, (2*k-1)^m * x^k +x*O(x^n))^m/m ) ), n)}
for(n=0, 30, print1(a(n), ", "))

{A060187(n, k) = sum(j=1, k, (-1)^(k-j) * binomial(n, k-j) * (2*j-1)^(n-1))}
{a(n) = my(A=1, Oxn=x*O(x^n));
A = exp( sum(m=1,n+1, sum(k=0, m, A060187(m+1, k+1)*x^k)^m /(1-x +Oxn)^(m^2+m) * x^m/m ) );
polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f.: exp( Sum_{n>=1} [ Sum_{k=0..n} A060187(n+1,k+1) * x^k ]^n / (1-x)^(n^2+n) * x^n/n ), where A060187 are the Eulerian numbers of type B.

A156631 G.f.: A(x) = Sum_{n>=0} ( Sum_{k>=1} (2^n2^kx)^k/k )^n / n!, a power series in x with integer coefficients.

Original entry on oeis.org

1, 4, 64, 3072, 466944, 283115520, 814634500096, 10734635101192192, 601470215201514061824, 138785509787119430915850240, 130376354694095237162362352959488

Offset: 0

Paul D. Hanna, Feb 12 2009

nonn

Compare to these dual g.f.s:
Sum_{n>=0} ( Sum_{k>=1} (2^n*x)^k/k )^n/n! (A060690);
Sum_{n>=0} ( Sum_{k>=1} (2^k*x)^k/k )^n/n! (A155200);
which, when expanded as power series in x, have only integer coefficients.

			G.f.: A(x) = 1 + 4*x + 64*x^2 + 3072*x^3 + 466944*x^4 + 283115520*x^5 + ...
From _Paul D. Hanna_, Mar 10 2009: (Start)
Let B(x) be the g.f. of A155200:
B(x) = 1 + 2*x + 10*x^2 + 188*x^3 + 16774*x^4 + 6745436*x^5 + ...
then a(n) is the coefficient of x^n in B(x)^(2^n):
B(x)^(2^0): [(1),2,10,188,16774,6745436,11466849412,...];
B(x)^(2^1): [1,(4),24,416,34400,13561728,22961051392,...];
B(x)^(2^2): [1,8,(64),1024,72704,27418624,46032420864,...];
B(x)^(2^3): [1,16,192,(3072),165888,56131584,92513894400,...];
B(x)^(2^4): [1,32,640,12288,(466944),118751232,186897137664,...];
B(x)^(2^5): [1,64,2304,65536,2129920,(283115520),382143037440,...];
B(x)^(2^6): [1,128,8704,425984,17956864,1140850688,(814634500096),...];
the terms along the diagonal (in parentheses) form this sequence. (End)

Cf. A156630, A060690, A155200.

{a(n)=polcoeff(sum(j=0,n,sum(k=1, n, (2^(j+k)*x)^k/k+x*O(x^n))^j/j!),n)}

/* a(n) = [x^n] B(x)^(2^n) where B(x) is g.f. of A155200: */ {a(n)=polcoeff(exp( 2^n*sum(k=1,n, 2^(k^2)*x^k/k)+x*O(x^n)), n)} \\ Paul D. Hanna, Mar 11 2009

a(n) = [x^n] B(x)^(2^n) where B(x) = exp(Sum_{n>=1} 2^(n^2)*x^n/n) is the g.f. of A155200. - Paul D. Hanna, Mar 10 2009

A158096 G.f.: A(x) = exp( Sum_{n>=1} x^n/n * 2^(n^2)/(1 + 2^(n^2)*x^n) ).

Original entry on oeis.org

1, 2, 6, 188, 16614, 6744492, 11466697660, 80444371592472, 2306003921102413254, 268654794307394089145676, 126765597337037378441876059252, 241678070947171631269022075304755208, 1858395916567280733577643964109494506976348, 57560683587055569906379529978030563771752589955832

Offset: 0

Paul D. Hanna, Mar 26 2009

nonn

Compare to g.f. of A010054:

exp( Sum_{n>=1} x^n/(1 + x^n)/n ) = 1 + x + x^3 + x^6 + x^10 +...

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 188*x^3 + 16614*x^4 + 6744492*x^5 +...
where
log(A(x)) = 2/(1 + 2*x)*x + 2^4/(1 + 2^4*x^2)*x^2/2 + 2^9/(1 + 2^9*x^3)*x^3/3 + 2^16/(1 + 2^16*x^4)*x^4/4 + 2^25/(1 + 2^25*x^5)*x^5/5 +...
Explicitly,
log(A(x)) = 2*x + 8*x^2/2 + 536*x^3/3 + 64960*x^4/4 + 33554592*x^5/5 + 68718964352*x^6/6 + 562949953422208*x^7/7 +...+ A262826(n)*x^n/n +...

Cf. A262826 (log), A262825, A158097, A155200.

{a(n)=if(n==0,1,polcoeff(exp(sum(k=1,n, x^k/k * 2^(k^2)/(1 + 2^(k^2)*x^k +x*O(x^n)))),n))}
for(n=0,20,print1(a(n),", "))

{a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, -(-1)^d * 2^(m^2/d) * d) ) +x*O(x^n)), n)}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Oct 02 2015

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} -(-1)^d * 2^(n^2/d) * d ). - Paul D. Hanna, Oct 02 2015

Logarithmic derivative equals A262826.

A162580 G.f.: A(x) = exp( 2Sum_{n>=1} 2^[A007814(n)^2] x^n/n ), where A007814(n) = exponent of highest power of 2 dividing n.

Original entry on oeis.org

1, 2, 4, 6, 16, 26, 44, 62, 240, 418, 756, 1094, 2544, 3994, 6556, 9118, 32352, 55586, 99492, 143398, 330000, 516602, 845900, 1175198, 3452112, 5729026, 9953556, 14178086, 31076592, 47975098, 77547580, 107120062, 298608832, 490097602

Offset: 0

Paul D. Hanna, Jul 06 2009

nonn

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 6*x^3 + 16*x^4 + 26*x^5 + 44*x^6 + ...
log(A(x))/2 = 2^0*x + 2^1*x^2 + 2^0*x^3/3 + 2^4*x^4/4 + 2^0*x^5/5 + 2^1*x^6/6 + 2^0*x^7/7 + 2^9*x^8/8 + ... + 2^[A007814(n)^2]*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A155200, A007814, A000123, A162581, A162582.

nmax = 500; a[n_]:= SeriesCoefficient[Series[Exp[ Sum[2^(IntegerExponent[k, 2]^2 + 1)*q^k/k, {k, 1, nmax}]], {q,0,nmax}], n]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jul 04 2018 *)

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

A162582 G.f.: A(x) = exp( 2Sum_{n>=1} A006519(n)^n x^n/n ), where A006519(n) = highest power of 2 dividing n.

Original entry on oeis.org

1, 2, 6, 10, 146, 282, 826, 1370, 4204986, 8408602, 25223066, 42037530, 615687706, 1189337882, 3483938586, 5778539290, 2305851850537847066, 4611703695297154842, 13835111074334385946, 23058518453371617050

Offset: 0

Paul D. Hanna, Jul 06 2009

nonn

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 146*x^4 + 282*x^5 + 826*x^6 + ...
log(A(x))/2 = 2^0*x + 2^2*x^2 + 2^0*x^3/3 + 2^8*x^4/4 + 2^0*x^5/5 + 2^6*x^6/6 + 2^0*x^7/7 + 2^24*x^8/8 + ... + A006519(n)^n*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A162580, A162581, A155200, A006519, A000123.

nmax = 200; a[n_]:= SeriesCoefficient[Series[Exp[ Sum[2^(k*IntegerExponent[k, 2] + 1)*q^k/k, {k, 1, nmax}]], {q,0,nmax}], n]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jul 04 2018 *)

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

cp's OEIS Frontend

A166168 G.f.: exp( Sum_{n>=1} Lucas(n^2)*x^n/n ) where Lucas(n) = A000204(n).

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A156171 G.f.: A(x) = exp( Sum_{n>=1} x^n/(1 - 2^n*x)^n / n ), a power series in x with integer coefficients.

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A165937 G.f.: A(x) = exp( Sum_{n>=1} A002203(n^2)*x^n/n ).

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A202518 G.f. satisfies: A(x) = exp( Sum_{n>=1} (2^n - A(x))^n * x^n/n ).

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A211892 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n^2) * x^n/n ), where Jacobsthal(n) = A001045(n).

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A292500 G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} (2*k-1)^n * x^k ]^n / n ).

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A156631 G.f.: A(x) = Sum_{n>=0} ( Sum_{k>=1} (2^n*2^k*x)^k/k )^n / n!, a power series in x with integer coefficients.

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A158096 G.f.: A(x) = exp( Sum_{n>=1} x^n/n * 2^(n^2)/(1 + 2^(n^2)*x^n) ).

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A292500 G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} (2k-1)^n x^k ]^n / n ).

A156631 G.f.: A(x) = Sum_{n>=0} ( Sum_{k>=1} (2^n2^kx)^k/k )^n / n!, a power series in x with integer coefficients.

A162580 G.f.: A(x) = exp( 2Sum_{n>=1} 2^[A007814(n)^2] x^n/n ), where A007814(n) = exponent of highest power of 2 dividing n.

A162582 G.f.: A(x) = exp( 2Sum_{n>=1} A006519(n)^n x^n/n ), where A006519(n) = highest power of 2 dividing n.