A158097
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, 14, 204, 16982, 6746636, 11467009772, 80444425963128, 2306004014991374374, 268654794950955551450892, 126765597355485863873077402788, 241678070949320869650125781001909864
Offset: 0
G.f.: A(x) = 1 + 2*x + 14*x^2 + 204*x^3 + 16982*x^4 + 6746636*x^5 +...
log(A(x)) = 2*x + 24*x^2/2 + 536*x^3/3 + 66112*x^4/4 + 33554592*x^5/5 +...
log(A(x)) = 2*x/(1-2*x) + 2^4*x^2/(1-2^4*x^2)/2 + 2^9*x^3/(1-2^9*x^3)/3 +...
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{a(n)=if(n==0,1,polcoeff(exp(sum(k=1,n,(2^k*x)^k/(1-(2^k*x)^k +x*O(x^n))/k)),n))}
for(n=0, 15, print1(a(n), ", "))
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{a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, 2^(m*d) * m/d) ) +x*O(x^n)), n)}
for(n=0, 15, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 30 2015
A262825
G.f.: exp( Sum_{n>=1} x^n/n * 3^(n^2)/(1 + 3^(n^2)*x^n) ).
Original entry on oeis.org
1, 3, 36, 6687, 10778724, 169490235888, 25016281203451185, 34185693515006540036988, 429210580094320740939492003864, 49269611092414637899756996954236224964, 51537752221009977969797967261567868189156062717, 490093718313094497196906721743434604298698182091562669412, 42294065506171581216310316433524563489152717911911865589940031127588
Offset: 0
G.f.: A(x) = 1 + 3*x + 36*x^2 + 6687*x^3 + 10778724*x^4 + 169490235888*x^5 +...
where
log(A(x)) = 3*x/(1 + 3*x) + 3^4/(1 + 3^4*x^2)*x^2/2 + 3^9/(1 + 3^9*x^3)*x^3/3 + 3^16/(1 + 3^16*x^4)*x^4/4 + 3^25/(1 + 3^25*x^5)*x^5/5 +...
Explicitly,
log(A(x)) = 3*x + 63*x^2/2 + 19764*x^3/3 + 43033275*x^4/4 + 847288610658*x^5/5 + 150094634523748092*x^6/6 + + 239299329230617529605392*x^7/7 +...
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{a(n)=if(n==0, 1, polcoeff(exp(sum(k=1, n, x^k/k * 3^(k^2)/(1 + 3^(k^2)*x^k +x*O(x^n)))), n))}
for(n=0, 20, print1(a(n), ", "))
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{a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, -(-1)^d * 3^(m^2/d) * d) ) +x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))
A262826
a(n) = Sum_{d|n} -(-1)^d * 2^(n^2/d) * d.
Original entry on oeis.org
2, 8, 536, 64960, 33554592, 68718964352, 562949953422208, 18446744065119352832, 2417851639229258752070144, 1267650600228227149696894752768, 2658455991569831745807614120560711680, 22300745198530623141526273540526772167065600, 748288838313422294120286634350736906063837462110208, 100433627766186892221372630770688837357523572410678079422464
Offset: 1
L.g.f.: L(x) = 2*x + 8*x^2/2 + 536*x^3/3 + 64960*x^4/4 + 33554592*x^5/5 + ...
where
L(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 + ...
and
exp(L(x)) = 1 + 2*x + 6*x^2 + 188*x^3 + 16614*x^4 + 6744492*x^5 + 11466697660*x^6 + 80444371592472*x^7 + ...+ A158096(n)*x^n + ...
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a[n_] := DivisorSum[n, -(-1)^# * 2^(n^2/#) * # &]; Array[a, 14] (* Amiram Eldar, Aug 24 2023 *)
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{a(n) = n*polcoeff(sum(k=1, n, x^k/k * 2^(k^2)/(1 + 2^(k^2)*x^k +x*O(x^n))), n)}
for(n=1,20,print1(a(n),", "))
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{a(n) = sumdiv(n, d, -(-1)^d * 2^(n^2/d) * d)}
for(n=1,20,print1(a(n),", "))
Showing 1-3 of 3 results.
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