A360732
Expansion of Sum_{k>0} (k * x * (1 + (k * x)^k))^k.
Original entry on oeis.org
1, 5, 27, 288, 3125, 48907, 823543, 17039360, 387479538, 10048828125, 285311670611, 8929262337009, 302875106592253, 11116754387067959, 437894195556640625, 18448995890703106048, 827240261886336764177, 39347760450413560593753
Offset: 1
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a[n_] := DivisorSum[n, #^n * Binomial[#, n/# - 1] &]; Array[a, 20] (* Amiram Eldar, Aug 09 2023 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x*(1+(k*x)^k))^k))
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a(n) = sumdiv(n, d, d^n*binomial(d, n/d-1));
A360726
Expansion of Sum_{k>0} (k * x * (1 + x^k))^k.
Original entry on oeis.org
1, 5, 27, 264, 3125, 46741, 823543, 16778240, 387420570, 10000015625, 285311670611, 8916100729755, 302875106592253, 11112006831322817, 437893890380890625, 18446744073843770368, 827240261886336764177, 39346408075300025059665
Offset: 1
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a[n_] := DivisorSum[n, #^# * Binomial[#, n/# - 1] &]; Array[a, 20] (* Amiram Eldar, Aug 02 2023 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x*(1+x^k))^k))
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a(n) = sumdiv(n, d, d^d*binomial(d, n/d-1));
A360759
a(n) = Sum_{d|n} d^(d+n/d) * binomial(d,n/d).
Original entry on oeis.org
1, 16, 243, 4112, 78125, 1680345, 40353607, 1073766400, 31381060338, 1000000781250, 34522712143931, 1283918489808640, 51185893014090757, 2177953338656796883, 98526125335697265625, 4722366482899710050304, 239072435685151324847153
Offset: 1
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a[n_] := DivisorSum[n, #^(# + n/#) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Aug 02 2023 *)
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a(n) = sumdiv(n, d, d^(d+n/d)*binomial(d, n/d));
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^k*((1+k*x^k)^k-1)))
A360770
Expansion of Sum_{k>0} (x * (k + x^k))^k.
Original entry on oeis.org
1, 5, 27, 260, 3125, 46684, 823543, 16777472, 387420498, 10000003125, 285311670611, 8916100495009, 302875106592253, 11112006826381559, 437893890380860625, 18446744073726328848, 827240261886336764177, 39346408075296925015353
Offset: 1
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a[n_] := DivisorSum[n, #^(# - n/# + 1) * Binomial[#, n/# - 1] &]; Array[a, 20] (* Amiram Eldar, Aug 02 2023 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (x*(k+x^k))^k))
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a(n) = sumdiv(n, d, d^(d-n/d+1)*binomial(d, n/d-1));
Showing 1-4 of 4 results.