A209397
L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} a(k)*x^(n*k)/k ).
Original entry on oeis.org
1, 3, 7, 19, 46, 129, 337, 939, 2581, 7238, 20263, 57337, 162319, 461961, 1317217, 3767035, 10792400, 30983565, 89084845, 256531814, 739658815, 2135234247, 6170505666, 17849457873, 51679366171, 149750711581, 434260829464, 1260198317509, 3659410074933
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 46*x^5/5 + 129*x^6/6 +...
Let G(x) be the g.f. of A000081, then
exp(L(x)) = G(x)/x where G(x) = x*exp( Sum_{n>=1} G(x^n)/n ) begins:
G(x) = x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 20*x^6 + 48*x^7 + 115*x^8 + 286*x^9 + 719*x^10 + 1842*x^11 + 4766*x^12 + 12486*x^13 + 32973*x^14 +...
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{a(n)=local(L=vector(n, i, 1)); for(i=1, n, L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, n\m, L[k]*x^(m*k)/k)+x*O(x^n)))))); L[n]}
for(n=1,30,print1(a(n),","))
A203254
G.f.: A(x) = exp( Sum_{n>=1} G_n(x^n)*x^n/n ) such that G_n(x^n) = Product_{k=0..n-1} A(u^k*x) where u is an n-th root of unity.
Original entry on oeis.org
1, 1, 2, 4, 10, 22, 62, 146, 422, 1084, 3160, 8064, 25190, 65204, 198652, 545790, 1680122, 4495548, 14352768, 38665478, 122530052, 343978146, 1072985932, 2947659006, 9662067644, 26573691092, 84395544446, 241295995524, 769819399580, 2140972333774, 7039688293036, 19579468840840
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 22*x^5 + 62*x^6 + 146*x^7 +...
G.f.: A(x) = exp( Sum_{n>=1} A203253(n)*x^n/n ),
where A(x) = exp( Sum_{n>=1} G_n(x^n)*x^n/n )
and G_n(x) = exp( Sum_{k>=1} A203253(n*k)*x^k/k ), which begin:
G_1(x) = A(x);
G_2(x) = 1 + 3*x + 16*x^2 + 104*x^3 + 724*x^4 + 5428*x^5 + 44080*x^6 +...;
G_3(x) = 1 + 7*x + 122*x^2 + 2128*x^3 + 52330*x^4 + 1109386*x^5 +...;
G_4(x) = 1 + 23*x + 1080*x^2 + 67944*x^3 + 4595792*x^4 +...;
G_5(x) = 1 + 51*x + 8582*x^2 + 1482524*x^3 + 355949360*x^4 +...;
G_6(x) = 1 + 195*x + 89752*x^2 + 53146664*x^3 + 36695632888*x^4 +...;
G_7(x) = 1 + 435*x + 705756*x^2 + 1208493276*x^3 +...;
G_8(x) = 1 + 1631*x + 7232560*x^2 + 44157620896*x^3 ...;
...
Also, G_n(x^n) = Product_{k=0..n-1} A(u^k*x) where u = n-th root of unity:
G_2(x^2) = A(x)*A(-x);
G_3(x^3) = A(x)*A(u*x)*A(u^2*x) where u = exp(2*Pi*I/3);
G_4(x^4) = A(x)*A(I*x)*A(I^2*x)*A(I^3*x) where I^2 = -1;
...
The logarithmic derivative of this sequence yields A203253:
A203253 = [1,3,7,23,51,195,435,1631,4165,14563,34761,141479,...].
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{a(n)=local(L=vector(n,i,1));for(i=1,n,L=Vec(deriv(sum(m=1,n,x^m/m*exp(sum(k=1,floor(n/m),L[m*k]*x^(m*k)/k)+x*O(x^n))))));polcoeff(exp(x*Ser(vector(n,m,L[m]/m))),n)}
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{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(m=1,n,x^m/m*round(prod(k=0,m-1,subst(A,x,exp(2*Pi*I*k/m)*x+x*O(x^n)))))));polcoeff(A,n)}
A203265
L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} 2*a(n*k)*x^(n*k)/k ).
Original entry on oeis.org
1, 5, 22, 125, 576, 3554, 16843, 103917, 521338, 3189600, 15813205, 101516930, 501568809, 3154939135, 16288999167, 101770328205, 513944896547, 3322082384450, 16707380500562, 106553006536680, 554390049927421, 3479202589748077, 17774723219041838
Offset: 1
L.g.f.: L(x) = x + 5*x^2/2 + 22*x^3/3 + 125*x^4/4 + 576*x^5/5 + 3554*x^6/6 +...
L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} G_n(x^n)^2*x^n/n
where G_n(x) = exp( Sum_{k>=1} a(n*k)*x^k/k ), which begin:
G_1(x) = 1 + x + 3*x^2 + 10*x^3 + 43*x^4 + 172*x^5 + 852*x^6 +...
G_2(x) = 1 + 5*x + 75*x^2 + 1518*x^3 + 34663*x^4 + 867760*x^5 +...;
G_3(x) = 1 + 22*x + 2019*x^2 + 214648*x^3 + 31221037*x^4 +...;
G_4(x) = 1 + 125*x + 59771*x^2 + 40659310*x^3 + 31438395303*x^4 +...;
G_5(x) = 1 + 576*x + 1760688*x^2 + 6380121685*x^3 +...;
G_6(x) = 1 + 3554*x + 57073923*x^2 + 1295238092004*x^3 +...;
G_7(x) = 1 + 16843*x + 1719312892*x^2 + 212162358939394*x^3 +...;
G_8(x) = 1 + 103917*x + 56284535547*x^2 + 44125115136389518*x^3 +...; ...
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{a(n)=local(L=vector(n, i, 1)); for(i=1, n, L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor(n/m), 2*L[m*k]*x^(m*k)/k)+x*O(x^n)))))); L[n]}
A203267
L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} 3*a(n*k)*x^(n*k)/k ).
Original entry on oeis.org
1, 7, 46, 371, 2611, 22444, 163010, 1414763, 10666423, 92901977, 700765693, 6267591344, 47400875250, 421269688378, 3261487427911, 28956966303371, 222519855315655, 2011947117233155, 15451470070634425, 138876292766145541, 1085821838608348370, 9706788507990083429
Offset: 1
L.g.f.: L(x) = x + 7*x^2/2 + 46*x^3/3 + 371*x^4/4 + 2611*x^5/5 +...
L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} G_n(x^n)^3*x^n/n
where G_n(x) = exp( Sum_{k>=1} a(n*k)*x^k/k ), which begin:
G_1(x) = 1 + x + 4*x^2 + 19*x^3 + 116*x^4 + 683*x^5 + 4818*x^6 +...
G_2(x) = 1 + 7*x + 210*x^2 + 8837*x^3 + 427910*x^4 + 22758491*x^5 +...;
G_3(x) = 1 + 46*x + 12280*x^2 + 4087909*x^3 + 1805475734*x^4 +...;
G_4(x) = 1 + 371*x + 776202*x^2 + 2360146453*x^3 +...;
G_5(x) = 1 + 2611*x + 49859649*x^2 + 1211412677799*x^3 +...;
G_6(x) = 1 + 22444*x + 3385662240*x^2 + 742868246890817*x^3 +...;
G_7(x) = 1 + 163010*x + 223920974239*x^2 + 396998122840515180*x^3 +...; ...
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{a(n)=local(L=vector(n, i, 1)); for(i=1, n, L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor(n/m), 3*L[m*k]*x^(m*k)/k)+x*O(x^n)))))); L[n]}
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