A308396
Expansion of e.g.f. exp(-Sum_{k>=1} x^(k^2)/k^2).
Original entry on oeis.org
1, -1, 1, -1, -5, 29, -89, 209, 841, -50905, 458641, -2423521, 8243731, 158742869, -2450634185, 18519809489, -1402926535919, 21355930009679, -139305034406879, 306503668195775, 40578438892908331, -816475138658703091, 6941097158619626311, -24787202385366731311
Offset: 0
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nmax = 23; CoefficientList[Series[Exp[-Sum[x^(k^2)/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[(1 - x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
A308398
Expansion of e.g.f. exp(Sum_{k>=1} x^(k^2)*(x^(k^2) - 1)/k^2).
Original entry on oeis.org
1, -1, 3, -7, 19, -51, 61, 167, 6777, -107929, 1650691, -17839911, 157217083, -1229269627, 6185945949, -3251776921, -1151787785999, 10138302541647, 532690324952707, -14122245788830279, 443912721023736291, -7480012715591067331, 115775303074594208893, -1392396864130912381017
Offset: 0
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nmax = 23; CoefficientList[Series[Exp[Sum[x^(k^2) (x^(k^2) - 1)/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[1/(1 + x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
A305620
Expansion of e.g.f. log(1 + Sum_{k>=1} x^(k^2)/k^2).
Original entry on oeis.org
1, -1, 2, 0, -6, 60, -540, 3780, 12600, -199080, 3074400, -45738000, 511434000, -5621616000, 55394539200, 960323364000, -24001273296000, 498178528848000, -9994137465312000, 156104172544320000, -2076607873660320000, 18061446353670720000, 206725394268993600000
Offset: 1
E.g.f.: A(x) = x - x^2/2! + 2*x^3/3! - 6*x^5/5! + 60*x^6/6! - 540*x^7/7! + ...
exp(A(x)) = 1 + x + x^4/4 + x^9/9 + x^16/16 + ... + x^A000290(k)/A000290(k) + ...
exp(exp(A(x))-1) = 1 + x + x^2/2! + x^3/3! + 7*x^4/4! + 31*x^5/5! + ... + A205801(k)*x^k/k! + ... = Product_{j>=1} 1/(1 - x^j)^(A008836(j)/j).
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N:= 50: # for a(1)..a(N)
g:= log(1 + add(x^(k^2)/k^2,k=1..floor(sqrt(N)))):
S:= series(g,x,N+1):
seq(coeff(S,x,n)*n!,n=1..N); # Robert Israel, Jun 07 2018
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nmax = 23; Rest[CoefficientList[Series[Log[1 + Sum[x^k^2/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 23; Rest[CoefficientList[Series[Log[1 + Log[Product[1/(1 - x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}]]], {x, 0, nmax}], x] Range[0, nmax]!]
a[n_] := a[n] = Boole[IntegerQ[n^(1/2)]] (n - 1)! - Sum[k Binomial[n, k] Boole[IntegerQ[(n - k)^(1/2)]] (n - k - 1)! a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 23}]
A308381
Expansion of e.g.f. exp(Sum_{k>=1} x^(k^2)*(2 + x^(k^2))/(2*k^2)).
Original entry on oeis.org
1, 1, 2, 4, 16, 56, 256, 1072, 11264, 119296, 1075456, 9088256, 85292032, 894690304, 8968964096, 90882789376, 2409397682176, 40515889528832, 1051789297844224, 16251803853193216, 302342408330018816, 4444559976664662016, 84010278329827459072, 1289319649553742823424
Offset: 0
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nmax = 23; CoefficientList[Series[Exp[Sum[x^(k^2) (2 + x^(k^2))/(2 k^2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[1/(1 - x^(2 k - 1))^(LiouvilleLambda[2 k - 1]/(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!