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;
A356389
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) ) /k.
Original entry on oeis.org
1, 2, 10, 34, 218, 1308, 10596, 74688, 793152, 7931520, 94504320, 1054218240, 14662840320, 205279764480, 3427909632000, 50923531008000, 907545606912000, 16335820924416000, 323185344975360000, 6220416698689536000, 140360358705186816000, 3087927891514109952000
Offset: 1
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Table[n! * Sum[Sum[-(-1)^(k/d), {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 25}] (* Vaclav Kotesovec, Aug 07 2022 *)
Table[n! * Sum[(2*DivisorSigma[0, 2*k] - 3*DivisorSigma[0, k])/k, {k, 1, n}], {n, 1, 25}] (* Vaclav Kotesovec, Aug 07 2022 *)
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a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/k);
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, log(1+x^k)/k)/(1-x)))
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;
A356402
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k!) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 3, 16, 86, 626, 5267, 50793, 543279, 6544805, 86503762, 1242678141, 19259416827, 321457169151, 5736414618209, 108931865485750, 2191495621647324, 46604972526167314, 1043844453093239627, 24555321244430950299, 605239630722584461955, 15600222966916650541099
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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a356401(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(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, a356401(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356564
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k) )^x.
Original entry on oeis.org
1, 0, 2, 0, 28, -30, 888, -1260, 51728, -196560, 5293080, -22286880, 710229408, -4851269280, 138348035616, -1091188098000, 36482139114240, -379928382462720, 11812558481332992, -137793570801143040, 4609972759421554560, -67292912045817561600
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))^x))
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a048272(n) = sumdiv(n, d, (-1)^(n/d+1));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a048272(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
A354503
Expansion of e.g.f. ( Product_{k>0} (1 + x^k)^(1/k) )^exp(x).
Original entry on oeis.org
1, 1, 3, 14, 67, 424, 3093, 26060, 233917, 2427224, 27565317, 339002146, 4450167269, 63343680802, 964189902141, 15769859929260, 270255218753593, 4913097747513800, 94513145955643993, 1904990351069631390, 40153307898034641361, 893402292594225679438
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))^exp(x)))
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a354506(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/(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, a354506(j)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-6 of 6 results.