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.
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
Keywords
Examples
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,...].
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..100
Programs
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PARI
{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)} -
PARI
{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)}