A203253
L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} a(n*k)*x^(n*k)/k ).
Original entry on oeis.org
1, 3, 7, 23, 51, 195, 435, 1631, 4165, 14563, 34761, 141479, 327471, 1222287, 3267177, 11804959, 28562075, 114349947, 272702593, 1056583023, 2786781123, 9966908779, 24678676437, 101422669199, 243331437901, 915276550503, 2464145600011, 9064045943983, 22324762587821
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 51*x^5/5 + 195*x^6/6 +...
L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} G_n(x^n)*x^n/n
where G_n(x) = exp( Sum_{k>=1} a(n*k)*x^k/k ), which begin:
G_1(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 22*x^5 + 62*x^6 + 146*x^7 +...
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 ...; ...
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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))))));L[n]}
A203266
G.f.: A(x) = exp( Sum_{n>=1} G_n(x^n)^2 * 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, 3, 10, 43, 172, 852, 3719, 19290, 90469, 481825, 2295973, 12812880, 62122518, 346770241, 1744884177, 9830723932, 49268101457, 285020577850, 1438021808125, 8300553742452, 42955378420263, 247456097485363, 1271978597532857, 7520668963944205, 38700952324615078
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 43*x^4 + 172*x^5 + 852*x^6 + 3719*x^7 +...
G.f.: A(x) = exp( Sum_{n>=1} A203265(n) * x^n/n ),
where A(x) = exp( Sum_{n>=1} G_n(x^n)^2 * x^n/n )
and G_n(x) = exp( Sum_{k>=1} A203265(n*k)*x^k/k ), which begin:
G_1(x) = A(x);
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 +...;
...
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(u*x)*A(u^2*x)*A(u^3*x) where u = exp(2*Pi*I/4);
...
The logarithmic derivative of this sequence yields A203265:
A203265 = [1,5,22,125,576,3554,16843,103917,521338,3189600,...].
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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)))))); 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^2, x, exp(2*Pi*I*k/m)*x+x*O(x^n))))))); polcoeff(A, n)}
A203268
G.f.: A(x) = exp( Sum_{n>=1} G_n(x^n)^3 * 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, 4, 19, 116, 683, 4818, 31126, 232058, 1598611, 12315375, 86887285, 695017086, 4999457900, 40344295044, 298468091712, 2434392979661, 18077507384936, 150454415661096, 1125745880242406, 9386869540033292, 71518155964958242, 597727034006054509
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 19*x^3 + 116*x^4 + 683*x^5 + 4818*x^6 +...
G.f.: A(x) = exp( Sum_{n>=1} A203267(n) * x^n/n ),
where A(x) = exp( Sum_{n>=1} G_n(x^n)^3 * x^n/n )
and G_n(x) = exp( Sum_{k>=1} A203267(n*k)*x^k/k ), which begin:
G_1(x) = A(x);
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 +...;
G_8(x) = 1 + 1414763*x + 15479260324770*x^2 + 249608398400792533605*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(u*x)*A(u^2*x)*A(u^3*x) where u = exp(2*Pi*I/4);
...
The logarithmic derivative of this sequence yields A203267:
A203267 = [1,7,46,371,2611,22444,163010,1414763,10666423,...].
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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)))))); 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^3, x, exp(2*Pi*I*k/m)*x+x*O(x^n))))))); polcoeff(A, n)}
Showing 1-3 of 3 results.
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