A346547
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^exp(x).
Original entry on oeis.org
1, 1, 6, 36, 282, 2575, 28075, 340809, 4657996, 69874305, 1145441713, 20279904337, 386803154474, 7874727448757, 170678885319787, 3919163707551187, 95029714996046680, 2424604353738271201, 64940619086990938317, 1820746123923294245293, 53328181409328560026038
Offset: 0
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nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^Exp[x], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Exp[x] Sum[DivisorSigma[1, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002745[n_] := Sum[Binomial[n, k] DivisorSigma[1, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002745[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
A346545
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(exp(x)/k).
Original entry on oeis.org
1, 1, 5, 26, 175, 1384, 12933, 135050, 1582901, 20380208, 286577757, 4352682256, 71247772121, 1244923243966, 23166410620637, 456940648889070, 9521696033968393, 208851154175983608, 4812156417656806393, 116112764199821653284, 2928658457243240595901, 77042063713731887400418
Offset: 0
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nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(Exp[x]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Exp[x] Sum[DivisorSigma[0, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002746[n_] := Sum[Binomial[n, k] DivisorSigma[0, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002746[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
A346546
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(exp(-x)/k).
Original entry on oeis.org
1, 1, 1, 2, 15, 44, 485, 1854, 25781, 170288, 2477485, 12571140, 435748665, 2049818198, 64651106637, 628176476186, 18837010964105, 93248340364152, 6695745240354169, 33794005826851192, 2549048418922818525, 20209158430316698922, 1138228671555859916609
Offset: 0
-
nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(Exp[-x]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Exp[-x] Sum[DivisorSigma[0, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002744[n_] := Sum[(-1)^(n - k) Binomial[n, k] DivisorSigma[0, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002744[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
A347916
E.g.f.: Product_{k>=1} (1 + x^k)^exp(-x).
Original entry on oeis.org
1, 1, 0, 6, 6, 75, 1025, 1225, 43988, 471345, 5084387, 40870181, 866782774, 8473297261, 165871287465, 3934845305287, 23390789927784, 956832091069057, 21869141108144439, 269518811758178785, 8437830353620298346, 220696789738463945981, 3231280243441039496181, 125072102239522472394691
Offset: 0
-
N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+x^k)^exp(-x))))
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N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(-x)*sum(k=1, N, sigma(k>>valuation(k, 2))*x^k/k))))
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N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(-x)*sum(k=1, N, x^k/(k*(1-x^(2*k)))))))
Showing 1-4 of 4 results.
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