A340904
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * sigma_1(k) * a(n-k).
Original entry on oeis.org
1, 1, 5, 28, 225, 2206, 26174, 361278, 5704401, 101297701, 1998893240, 43386854622, 1027353587730, 26353742447280, 728030940612638, 21548668265211778, 680330296613877761, 22821706122361385354, 810587673640374442445, 30390159250481750866640
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] DivisorSigma[1, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[1/(1 - Sum[Sum[i x^(i j)/(i j)!, {j, 1, nmax}], {i, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, sigma(k)*x^k/k!)))) \\ Seiichi Manyama, Mar 29 2022
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a(n) = if(n==0, 1, sum(k=1, n, sigma(k)*binomial(n, k)*a(n-k))); \\ Seiichi Manyama, Mar 29 2022
A341505
E.g.f.: Product_{i>=1, j>=1} 1 / (1 - x^(i*j) / (i*j)!).
Original entry on oeis.org
1, 1, 4, 14, 77, 427, 3076, 23088, 205316, 1936275, 20611750, 233576818, 2909340750, 38527889389, 551372037898, 8364582709282, 135560933977809, 2320127265064403, 42072789623722518, 802547153889643250, 16118882845967168807, 339268639052195731063
Offset: 0
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nmax = 21; CoefficientList[Series[Product[1/(1 - x^k/k!)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[d DivisorSigma[0, d]/(d!)^(k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 21}]
A341506
E.g.f.: Product_{i>=1, j>=1} (1 + x^(i*j) / (i*j)!).
Original entry on oeis.org
1, 1, 2, 8, 17, 87, 366, 1514, 8770, 45585, 267586, 1612624, 11914416, 73215391, 522906754, 4364545708, 33150679697, 263662491935, 2151338992440, 20815916251604, 178593028936507, 1714283809331191, 15531842607259512, 158682350653110712, 1667852117293837230
Offset: 0
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nmax = 24; CoefficientList[Series[Product[(1 + x^k/k!)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, -(n - 1)! Sum[Sum[d DivisorSigma[0, d]/(-d!)^(k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 24}]
A352693
Expansion of e.g.f. 1 / (1 - Sum_{k>=1} sigma_2(k) * x^k/k!).
Original entry on oeis.org
1, 1, 7, 46, 455, 5406, 78172, 1312116, 25214479, 544777183, 13080808752, 345471545728, 9953804592152, 310687941345796, 10443489230611052, 376122782541917166, 14449157656748079247, 589772212576633845886, 25488817336672959449725
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, sigma(k, 2)*x^k/k!))))
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a(n) = if(n==0, 1, sum(k=1, n, sigma(k, 2)*binomial(n, k)*a(n-k)));
A352841
Expansion of e.g.f. 1/(1 - Sum_{k>=1} sigma_k(k) * x^k/k!).
Original entry on oeis.org
1, 1, 7, 64, 851, 13906, 277972, 6466650, 172651643, 5186830537, 173327806752, 6373233407498, 255743444526584, 11119651415719744, 520752884139087852, 26132341317365562754, 1398900109763305183707, 79569524691656775423766
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, sigma(k, k)*x^k/k!))))
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a(n) = if(n==0, 1, sum(k=1, n, sigma(k, k)*binomial(n, k)*a(n-k)));
A353186
Expansion of e.g.f. 1/(1 - Sum_{k>=1} d(k) * x^k / k), where d(n) = number of divisors of n (A000005).
Original entry on oeis.org
1, 1, 4, 22, 170, 1588, 18236, 240840, 3662424, 62456136, 1185150768, 24714979584, 562659843984, 13870798275072, 368324715871680, 10478253239415552, 317975367247809408, 10252138622419702656, 349999438215928660992, 12612365665457524786944, 478414908509124826439424
Offset: 0
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d[k_] := d[k] = DivisorSigma[0, k]; a[0] = 1; a[n_] := a[n] = Sum[(k - 1)! * d[k] * Binomial[n, k] * a[n - k], {k, 1, n}]; Array[a, 21, 0] (* Amiram Eldar, Apr 30 2022 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, numdiv(k)*x^k/k))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (j-1)!*numdiv(j)*binomial(i, j)*v[i-j+1])); v;
Showing 1-6 of 6 results.