A157312
G.f.: A(x) = exp(Sum_{n>=1} A157311(n)*x^n/n) = Product_{n>=1} (1 + A157311(n-1)*x^n).
Original entry on oeis.org
1, 1, 1, 2, 5, 18, 84, 481, 3249, 25359, 224000, 2208441, 24019991, 285633470, 3685413373, 51271476627, 764944009086, 12182390286127, 206262410584138, 3699483818281188, 70067511789111404, 1397379232420943285
Offset: 0
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 18*x^5 + 84*x^6 +...
where both the exponential:
A(x) = exp(x + x^2/2 + 4*x^3/3 + 13*x^4/4 + 66*x^5/5 + 394*x^6/6 +...)
and the product:
A(x) = (1 + x)(1 + x^2)(1 + x^3)(1 + 4*x^4)(1 + 13*x^5)(1 + 66*x^6)*...
generate A(x) using the same coefficients (after initial term):
A157311=[1,1,1,4,13,66,394,2759,22005,198049,1979646,21776107,...].
A157313
G.f.: exp( Sum_{n>=1} a(n)*x^n/n ) = Product_{n>=1} 1/(1 - a(n-1)*x^n).
Original entry on oeis.org
1, 1, 3, 10, 43, 216, 1326, 9283, 74667, 672085, 6730098, 74031079, 888657130, 11552542691, 161747905609, 2426218982400, 38820193151115, 659943283568956, 11879029341157575, 225701557481993926, 4514035666639844778, 94794749015757064732, 2085484976583065409751
Offset: 0
Define G(x) by the exponential:
G(x) = exp(x + 3*x^2/2 + 10*x^3/3 + 43*x^4/4 + 216*x^5/5 + 1326*x^6/6 +...)
then 1/G(x) also equals the product:
1/G(x) = (1 - x)(1 - x^2)(1 - 3*x^3)(1 - 10*x^4)(1 - 43*x^5)(1 - 216*x^6)*...
where the coefficients in both expressions are the same (with offset)
and G(x) is the g.f. of A157314:
G(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 62*x^5 + 298*x^6 + 1700*x^7 +...
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a[0] = 1; a[n_] := a[n] = DivisorSum[n, #*a[#-1]^(n/#) &]; Array[a, 20, 0] (* Amiram Eldar, Aug 15 2023 *)
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{a(n)=if(n==0,1,sumdiv(n,d,if(d>=1&d<=n,d*(a(d-1))^(n/d))))}
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{a(n)=if(n==0, 1,n*polcoeff(1+sum(k=1,n,-log(1-a(k-1)*x^k +x*O(x^n))),n))}
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{a(n)=if(n==0, 1,n*polcoeff(sum(k=1,n,sum(j=1,n\k,a(k-1)^j*x^(k*j)/j)+x*O(x^n)),n))}
A318600
a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d+1)*d*a(d).
Original entry on oeis.org
1, 1, 1, 4, 13, 66, 394, 2759, 22053, 198481, 1984746, 21832207, 261986098, 3405819275, 47681467093, 715222006464, 11443552081333, 194540385382662, 3501726936689833, 66532811797106828, 1330656235940151698, 27943780954743188420, 614763181004328313035, 14139553163099551199806
Offset: 1
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f:= proc(n) option remember;
add((-1)^((n-1)/d+1)*d*procname(d), d = numtheory:-divisors(n-1))
end proc:
f(1):= 1:
map(f, [$1..30]); # Robert Israel, Aug 30 2018
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a[1] = 1; a[n_] := a[n] = Sum[(-1)^((n - 1)/d + 1) d a[d] , {d, Divisors[n - 1]}]; Table[a[n], {n, 24}]
Showing 1-3 of 3 results.