A294394
E.g.f.: exp(Sum_{n>=1} A000593(n) * x^n).
Original entry on oeis.org
1, 1, 3, 31, 145, 1641, 17731, 194503, 2676801, 40644145, 667689571, 11514903951, 227665389073, 4578990563161, 100913115588195, 2372334731747191, 57930324367791361, 1509398686720812513, 41341036374519788611, 1184009909077133031295
Offset: 0
E.g.f.: exp(Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n):
A294392 (k=0), this sequence (k=1),
A294395 (k=2).
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a[n_] := a[n] = If[n == 0, 1, Sum[k*Sum[-(-1)^d*k/d, {d, Divisors[k]}]*a[n - k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 07 2018 *)
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N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, d*(d%2))*x^k))))
A294395
E.g.f.: exp(Sum_{n>=1} A050999(n) * x^n).
Original entry on oeis.org
1, 1, 3, 67, 289, 5121, 71731, 861043, 18134817, 303946849, 6724342531, 146426154051, 3533373668353, 93259190078497, 2489644674735219, 75193364720030131, 2265438714279130561, 74716734198386887233, 2543592184722884351107, 90853513680763023292099
Offset: 0
E.g.f.: exp(Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n):
A294392 (k=0),
A294394 (k=1), this sequence (k=2).
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N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, d^2*(d%2))*x^k))))
A295794
Expansion of e.g.f. Product_{k>=1} exp(x^k/(1 + x^k)).
Original entry on oeis.org
1, 1, 1, 13, 25, 241, 2761, 14701, 153553, 1903105, 27877681, 263555821, 4788201001, 65083782193, 1040877257785, 24098794612621, 373918687272481, 7393663746307201, 164894196647876833, 3504497611085823565, 81863829346282866361, 2257321249626793901041, 49755091945025205954601
Offset: 0
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a:=series(mul(exp(x^k/(1+x^k)),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019
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nmax = 22; CoefficientList[Series[Product[Exp[x^k/(1 + x^k)], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[x D[Log[Product[(1 + x^k)^(1/k), {k, 1, nmax}]], x]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[-k Sum[(-1)^d, {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 22}]
Showing 1-3 of 3 results.