A356392
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 3, 17, 99, 769, 6877, 70769, 807321, 10366037, 145721531, 2226927405, 36741898267, 651709348653, 12352436747141, 249152882935829, 5320544034698353, 120008265471779529, 2850195632804141203, 71058458112629765449, 1855470903727083981651
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^(1/k))^(1/(1-x))))
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a356389(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356389(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356025
Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^(1/k!) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 5, 28, 206, 1786, 18347, 212745, 2773927, 39901109, 628298992, 10725440221, 197349522471, 3888090474399, 81659016005387, 1820049574958950, 42895622543757084, 1065460090285463634, 27811791343693345811, 760920657403831436463
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace((1/prod(k=1, N, (1-x^k)^(1/k!)))^(1/(1-x))))
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a356009(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)!)));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356009(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356337
Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^k )^(1/(1-x)).
Original entry on oeis.org
1, 1, 8, 63, 644, 7610, 107994, 1713726, 30671024, 603160344, 12974475240, 301879678320, 7561610279112, 202437968475288, 5769455216675136, 174234738889383480, 5556311629901103360, 186482786151757707840, 6568881383985687359424, 242221409390815100812224
Offset: 0
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With[{nn=20},CoefficientList[Series[Product[1/((1-x^k)^k)^(1/(1-x)),{k,nn}],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 06 2023 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace((1/prod(k=1, N, (1-x^k)^k))^(1/(1-x))))
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a356298(n) = n!*sum(k=1, n, sigma(k, 2)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356298(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A355064
Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^(1/k) )^x.
Original entry on oeis.org
1, 0, 2, 6, 28, 210, 1248, 13020, 102128, 1248912, 13457880, 176726880, 2362784928, 36609693120, 551337892896, 9588702417840, 171779733546240, 3230529997766400, 64714946343904512, 1371420774325866240, 29953522454811096960, 698447624328756610560
Offset: 0
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a[0] := a[0] = 1; a[1] := a[1] = 0;
a[n_] := a[n] = Sum[Factorial[k]*DivisorSigma[0, k - 1]/(k - 1)*Binomial[n - 1, k - 1]* a[n - k], {k, 2, n}];
Table[a[n], {n, 0, 50}] (* Sidney Cadot, Jan 05 2023 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^(1/k))^x))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*sigma(j-1, 0)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
A356335
Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 6, 39, 332, 3290, 38994, 517986, 7762880, 128029464, 2311675560, 45188359920, 952047539112, 21452758881528, 515073388373712, 13114579450948920, 352881761400606720, 10000259978380933440, 297654582665846499264, 9280441162956638320704
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace((1/prod(k=1, N, 1-x^k))^(1/(1-x))))
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a356010(n) = n!*sum(k=1, n, sigma(k)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356010(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356408
Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 5, 29, 216, 1919, 20012, 236977, 3145832, 46122546, 739703182, 12865212172, 241040899668, 4836265824740, 103410589256452, 2346358252787094, 56285005757022752, 1422783492250963296, 37790069818311971640, 1051924374853915254048
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k/k)^(1/(1-x))))
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a356406(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)^d)));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356406(j)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-6 of 6 results.