A321385
a(n) = Sum_{d|n} (-1)^(n/d+1)*d^d.
Original entry on oeis.org
1, 3, 28, 251, 3126, 46632, 823544, 16776955, 387420517, 9999996878, 285311670612, 8916100401824, 302875106592254, 11112006824734476, 437893890380862528, 18446744073692774139, 827240261886336764178, 39346408075296150201567, 1978419655660313589123980, 104857599999999989999997126
Offset: 1
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Table[Sum[(-1)^(n/d + 1) d^d, {d, Divisors[n]}], {n, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[k^k x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
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a(n) = sumdiv(n, d, (-1)^(n/d+1)*d^d); \\ Michel Marcus, Nov 09 2018
A299786
Expansion of Product_{k>=1} (1 + k^(k-1)*x^k).
Original entry on oeis.org
1, 1, 2, 11, 73, 707, 8547, 127379, 2237804, 45511484, 1049155214, 27060763974, 771662014455, 24109614539775, 818906748562249, 30044648617150066, 1184045057676213763, 49883902402848781573, 2237286132689496359239, 106426356238092171308928, 5352031894869594850387969
Offset: 0
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nmax = 20; CoefficientList[Series[Product[(1 + k^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^(k - k/d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]
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N=40; x='x+O('x^N); Vec(prod(k=1, N, 1+k^(k-1)*x^k)) \\ Seiichi Manyama, Aug 22 2020
A321388
Expansion of Product_{k>=1} (1 + x^k)^(k^(k-2)).
Original entry on oeis.org
1, 1, 1, 4, 19, 144, 1443, 18295, 280918, 5069651, 105147307, 2464296222, 64402891501, 1856989724951, 58560557062508, 2004999890781363, 74069439021212783, 2936703201134924845, 124383305232306494864, 5605027085651919547476, 267759074907470856179460, 13516676464234372267564939
Offset: 0
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m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 + x^k)^(k^(k-2)): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018
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a:=series(mul((1+x^k)^(k^(k-2)),k=1..100),x=0,22): seq(coeff(a,x,n),n=0..21); # Paolo P. Lava, Apr 02 2019
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nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(k^(k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]
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m=30; x='x+O('x^m); Vec(prod(k=1,m,(1+x^k)^(k^(k-2)))) \\ G. C. Greubel, Nov 09 2018
Showing 1-3 of 3 results.
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