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;
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;
A356393
Expansion of e.g.f. ( Product_{k>0} (1+x^k) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 4, 27, 188, 1730, 18234, 220206, 2958416, 44470296, 729675720, 13002636240, 249986061192, 5154030469848, 113360272804128, 2648908519611480, 65477559553098240, 1707034986277780800, 46798324479957887424, 1345365460101611611584
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1+x^k)^(1/(1-x))))
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a356390(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356390(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;
A354623
Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k) )^x.
Original entry on oeis.org
1, 0, 2, 9, 44, 390, 2754, 32760, 310064, 4244184, 54887400, 818615160, 12909921672, 225872515440, 4045885572624, 79360837887240, 1649832369335040, 35666417240193600, 822291935260976064, 19830352438530840960, 501144432316767688320, 13229590606682042436480
Offset: 0
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nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^x, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 17 2022 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^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)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-5 of 5 results.