A303344
Expansion of Product_{n>=1} ((1 + (n*x)^n)/(1 - (n*x)^n))^(1/n).
Original entry on oeis.org
1, 2, 6, 28, 182, 1640, 19220, 278224, 4809942, 96598622, 2208156512, 56580566908, 1605518324884, 49963000166616, 1691615823420800, 61897541544248720, 2433873670903995990, 102341746590575878628, 4582360425862350559350, 217661837260679635780356
Offset: 0
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nmax = 20; CoefficientList[Series[Product[((1 + (k*x)^k)/(1 - (k*x)^k))^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)
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N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+(k*x)^k)/(1-(k*x)^k))^(1/k)))
A308228
G.f.: x * Product_{k>=1} 1/(1 - k^k*x^k)^(a(k)/k).
Original entry on oeis.org
1, 1, 3, 30, 1956, 1224510, 9523018859, 1120383171258352, 2349614928773045360884, 101143220645945325750097689653, 101143220747088551095300901321325558554, 2623394662131051405254078144558922468191548124266
Offset: 1
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a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - k^k x^k)^(a[k]/k), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 12}]
a[n_] := a[n] = Sum[Sum[d^k a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]/(n - 1); a[1] = 1; Table[a[n], {n, 1, 12}]
A318968
Expansion of exp(Sum_{k>=1} ( Sum_{d|k, d odd} d^k ) * x^k/k).
Original entry on oeis.org
1, 1, 1, 10, 10, 635, 797, 118446, 124071, 43174194, 45404910, 25982930761, 26443958420, 23324558686914, 23640266984002, 29216576615057082, 29447535265299613, 48690644491136860817, 48980258924147884960, 104176334607664412086539, 104636388540330684649083, 278323070872780066332365486
Offset: 0
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nmax = 21; CoefficientList[Series[Exp[Sum[Sum[Mod[d, 2] d^k, {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 21; CoefficientList[Series[Product[1/(1 - (2 k - 1)^(2 k - 1) x^(2 k - 1))^(1/(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Mod[d, 2] d^k, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]
A318969
Expansion of exp(Sum_{k>=1} ( Sum_{p|k, p prime} p^k ) * x^k/k).
Original entry on oeis.org
1, 0, 2, 9, 6, 643, 182, 118953, 6019, 242630, 2243190, 25938251679, 78106516, 23349992199606, 288964822371, 46755212195033, 226472341461312, 48661337027901364945, 18066374340919781, 104224677113940850317679, 440728415311733637734, 208546898802899685866735, 972477473959172989443327
Offset: 0
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nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 22; CoefficientList[Series[Product[1/(1 - Prime[k]^Prime[k] x^Prime[k])^(1/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 22}]
A356590
Expansion of e.g.f. ( Product_{k>0} 1/(1 - (k * x)^k)^(1/k) )^exp(x).
Original entry on oeis.org
1, 1, 8, 96, 2382, 100035, 6995185, 699004551, 96910745876, 17476222963065, 4000562831147323, 1127335505294104887, 384099492016873956422, 155403154609857016567601, 73680868272553092728379865, 40444727351284600806487687057
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-(k*x)^k)^(1/k))^exp(x)))
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a356589(n) = n!*sum(k=1, n, sigma(k, k)/(k*(n-k)!));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356589(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A158265
G.f.: A(x) = exp( Sum_{n>=1} 2*sigma(n,n+1)*x^n/n ).
Original entry on oeis.org
1, 2, 11, 74, 697, 8002, 115158, 1949640, 38662510, 872245634, 22150393253, 623661939852, 19296665400632, 650198159192554, 23700604926216759, 928939297013294294, 38956230043045053042, 1740248411222193973416
Offset: 0
G.f.: A(x) = 1 + 2*x + 11*x^2 + 74*x^3 + 697*x^4 + 8002*x^5 +...
log(A(x)) = 2*x + 18*x^2/2 + 164*x^3/3 + 2114*x^4/4 + 31252*x^5/5 +...
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a(n)=polcoeff(exp(sum(m=1, n, 2*sigma(m, m+1)*x^m/m)+x*O(x^n)), n)
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