A356336
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, 29, 219, 1949, 20587, 245237, 3289577, 48670973, 788572541, 13849348105, 262283664739, 5317530185889, 114939490137235, 2636612228192969, 63955437488072593, 1634890446576454297, 43920715897460109205, 1236660724225711901749, 36412086992371220561771
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace((1/prod(k=1, N, (1-x^k)^(1/k)))^(1/(1-x))))
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a356297(n) = n!*sum(k=1, n, sigma(k, 0)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356297(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356394
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^k )^(1/(1-x)).
Original entry on oeis.org
1, 1, 6, 51, 452, 5210, 68514, 1032906, 17352320, 323948376, 6594052680, 145585638000, 3461441121192, 88092914635128, 2388119359650192, 68667743686492440, 2086307088847714560, 66762608893508354880, 2243693428523140377024, 78982154604162553529664
Offset: 0
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nmax = 20; CoefficientList[Series[Product[(1+x^k)^k, {k, 1, nmax}]^(1/(1-x)), {x, 0, nmax}], x] * Range[0,nmax]! (* Vaclav Kotesovec, Aug 07 2022 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^k)^(1/(1-x))))
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a356391(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356391(j)*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;
A353993
Expansion of e.g.f. ( Product_{k>0} 1/(1 - k * x^k) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 8, 63, 668, 7850, 115914, 1847286, 34031024, 682177464, 15049816200, 357564279600, 9212847784392, 252552128708568, 7395084613746816, 229412209982127480, 7524339637608261120, 259675490280634374720, 9418707076419411194304, 357606237255136232451264
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/(1-x))))
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a353992(n) = n!*sum(k=1, n, sumdiv(k, d, (k/d)^d/d));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a353992(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356554
Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^k )^x.
Original entry on oeis.org
1, 0, 2, 15, 92, 930, 8514, 116760, 1445744, 23020200, 373858920, 6756785640, 130982295432, 2751191997840, 61046788571664, 1445520760702200, 36387213668348160, 960383111961228480, 26780931923301572544, 781864626481646405760, 23925584882896903854720
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^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, 2)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
A356595
Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k)^k )^exp(x).
Original entry on oeis.org
1, 1, 8, 60, 582, 6555, 88585, 1333731, 22602020, 420261225, 8536210843, 187294058787, 4420961159582, 111409233290537, 2986570482052729, 84773698697674837, 2539347801355477960, 80003306259203052465, 2644032803825175398175, 91425359712959262036223
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)^exp(x)))
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a356600(n) = n!*sum(k=1, n, sigma(k, 2)/(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, a356600(j)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-6 of 6 results.