A110152
G.f.: A(x) = Product_{n>=1} 1/(1 - 2^n*x^n)^(2/2^n).
Original entry on oeis.org
1, 2, 6, 14, 36, 78, 192, 406, 942, 2018, 4512, 9450, 21178, 43950, 95532, 200398, 431356, 892518, 1917572, 3950614, 8410230, 17398466, 36648980, 75326754, 159199004, 326471706, 683028924, 1404145162, 2930071798, 5993625942
Offset: 0
G.f.: A(x) = 1 + 2*x + 6*x^2 + 14*x^3 + 36*x^4 + 78*x^5 +...
where
A(x) = 1/((1-2*x) * (1-4*x^2)^(1/2) * (1-8*x^3)^(1/4) * (1-16*x^4)^(1/8) *...).
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nmax = 30; CoefficientList[Series[Product[1/(1 - 2^k*x^k)^(2/2^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 18 2020 *)
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a(n)=polcoeff(prod(k=1,n,1/(1-2^k*x^k+x*O(x^n))^(2/2^k)),n)
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A090879(n) = sumdiv(n,d, d*2^(n-d))
a(n)=local(A);A=exp(sum(k=1,n,2*A090879(k)*x^k/k)+x*O(x^n));polcoeff(A,n)
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Jan 05 2014
A356539
a(n) = Sum_{d|n} d * 3^(n-d).
Original entry on oeis.org
1, 5, 12, 49, 86, 492, 736, 3977, 8757, 34030, 59060, 384924, 531454, 2672528, 6672552, 26093113, 43046738, 261646137, 387420508, 2181624374, 4682526672, 17435870644, 31381059632, 204908769276, 299863458511, 1412168408630, 3392641222200, 13912336721584
Offset: 1
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a[n_] := DivisorSum[n, # * 3^(n - #) &]; Array[a, 30] (* Amiram Eldar, Aug 11 2022 *)
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a(n) = sumdiv(n, d, d*3^(n-d));
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my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1-(3*x)^k)))
A356538
Expansion of e.g.f. Product_{k>0} 1/(1 - (2 * x)^k)^(1/2^k).
Original entry on oeis.org
1, 1, 5, 27, 249, 2085, 30645, 354375, 6542865, 108554985, 2330525925, 45331607475, 1288779532425, 28889867731725, 876160258298325, 25315531795929375, 860642393272286625, 26527678331237708625, 1063065483349950205125, 36393649136002135852875
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-(2*x)^k)^(1/2^k))))
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a090879(n) = sumdiv(n, d, d*2^(n-d));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, a090879(j)*v[i-j+1]/(i-j)!)); v;
A358660
a(n) = Sum_{d|n} d * (n/d)^(n-d).
Original entry on oeis.org
1, 4, 12, 76, 630, 7968, 117656, 2105416, 43048917, 1000781420, 25937424612, 743130116112, 23298085122494, 793742455829456, 29192926758107760, 1152930300766980112, 48661191875666868498, 2185915267189632382650, 104127350297911241532860
Offset: 1
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a[n_] := Total[Map[#*(n/#)^(n - #) &, Divisors[n]]];
Table[a[n],{n,1,100}]
a[n_] := DivisorSum[n, (n/#)^(n-#)*# &]; Array[a, 19] (* Amiram Eldar, Aug 27 2023 *)
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a(n) = sumdiv(n, d, d*(n/d)^(n-d));
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k-1)*x^k/(1-k^(k-1)*x^k)^2))
Showing 1-4 of 4 results.