A062796
Inverse Moebius transform of f(n) = n^n (A000312).
Original entry on oeis.org
1, 5, 28, 261, 3126, 46688, 823544, 16777477, 387420517, 10000003130, 285311670612, 8916100495200, 302875106592254, 11112006826381564, 437893890380862528, 18446744073726329093, 827240261886336764178, 39346408075296925042601, 1978419655660313589123980
Offset: 1
n=6: divisors = {1,2,3,6}; 1^1 + 2^2 + 3^3 + 6^6 = 1 + 4 + 27 + 46656 = 46688 = a(6).
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a[n_] := DivisorSum[n, #^# &]; Array[a, 19] (* Jean-François Alcover, Dec 23 2015 *)
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vector(17, n, sumdiv(n, d, d^d))
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{a(n)=polcoeff(sum(m=1,n,m^m*x^m/(1-x^m +x*O(x^n))),n)} \\ Paul D. Hanna, Oct 27 2009
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a(n) = sumdiv(n,d, d^d ); \\ Joerg Arndt, Apr 14 2013
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from sympy import divisors
def A062796(n): return sum(d**d for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022
A321387
Expansion of Product_{k>=1} (1 + x^k)^(k^(k-1)).
Original entry on oeis.org
1, 1, 2, 11, 74, 708, 8583, 127424, 2239965, 45514345, 1049365071, 27061132159, 771695223819, 24109698083919, 818914886275467, 30044684789498522, 1184048086192376822, 49883929845112421452, 2237287911899357657492, 106426388125032988691636, 5352033610656721914626572
Offset: 0
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m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1+x^k)^(k^(k-1)): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018
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a:=series(mul((1+x^k)^(k^(k-1)),k=1..100),x=0,21): seq(coeff(a,x,n),n=0..20); # Paolo P. Lava, Apr 02 2019
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nmax = 20; CoefficientList[Series[Product[(1 + x^k)^(k^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^d, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]
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seq(n)={Vec(exp(sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^d ) * x^k/k) + O(x*x^n)))} \\ Andrew Howroyd, Nov 09 2018
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m=30; x='x+O('x^m); Vec(prod(k=1,m,(1+x^k)^(k^(k-1)))) \\ G. C. Greubel, Nov 09 2018
A262842
G.f.: Product_{k>=1} (1 - x^k)^(-k^(k-2)).
Original entry on oeis.org
1, 1, 2, 5, 22, 150, 1469, 18452, 282426, 5088276, 105431378, 2469403421, 64508609896, 1859464257187, 58625171707730, 2006861834895431, 74128128916520263, 2938711927441481562, 124457492116819509679, 5607967808192795374759, 267883606645817181302028, 13522287374792657280601627, 719232962773594118661491002, 40204966834965724305054746851
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 22*x^4 + 150*x^5 + 1469*x^6 +...
where
A(x) = 1/((1-x)*(1-x^2)*(1-x^3)^3*(1-x^4)^16*(1-x^5)^125*(1-x^6)^1296*...)
also
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 67*x^4/4 + 626*x^5/5 + 7788*x^6/6 + 117650*x^7/7 + 2097219*x^8/8 + 43046731*x^9/9 + 1000000628*x^10/10 +...+ A262843(n)*x^n/n +...
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{a(n)=polcoeff(prod(k=1, n, (1 - x^k +x*O(x^n))^(-k^(k-2))), n)}
for(n=0,30,print1(a(n),", "))
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{a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, d^d)*x^m/m) +x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))
A283335
Expansion of exp( Sum_{n>=1} -A062796(n)/n*x^n ) in powers of x.
Original entry on oeis.org
1, -1, -2, -7, -54, -544, -7005, -108220, -1958263, -40629205, -951376217, -24826365255, -714568797261, -22491957589783, -768651303338761, -28344950796904518, -1121910285249842486, -47442295013058570884, -2134673855370621621400
Offset: 0
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A[n_] := Sum[d^d, {d, Divisors[n]}]; a[n_] := If[n==0, 1, -(1/n)*Sum[A[k]*a[n - k], {k, n}]]; Table[a[n], {n, 0, 18}] (* Indranil Ghosh, Mar 11 2017 *)
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a(n) = if(n==0, 1, -(1/n)*sum(k=1, n, sumdiv(k, d, d^d)*a(n - k)));
for(n=0, 18, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 11 2017
A323634
Expansion of Product_{k>=1} 1/(1 - k^(k-1)*x^k).
Original entry on oeis.org
1, 1, 3, 12, 80, 723, 8716, 128227, 2251086, 45647542, 1051845574, 27107414480, 772785074811, 24136982014698, 819697939365724, 30068912837398063, 1184872370227462528, 49914074776385885492, 2238476211786621770206, 106476394492364281869654, 5354276181476337307494676
Offset: 0
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a:=series(mul(1/(1-k^(k-1)*x^k),k=1..100),x=0,21): seq(coeff(a,x,n),n=0..20); # Paolo P. Lava, Apr 02 2019
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nmax = 20; CoefficientList[Series[Product[1/(1 - k^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[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(1/prod(k=1, N, 1-k^(k-1)*x^k)) \\ Seiichi Manyama, Aug 22 2020
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