A119470
G.f. A(x) equals the limit of the composition of functions (x+x^n); let F_1(x) = x, F_{n+1}(x) = F_n(x+x^(n+1)), then A(x) = limit F_n(x): A(x) = x o x+x^2 o x+x^3 o ... o x+x^n o...
Original entry on oeis.org
1, 1, 1, 3, 3, 7, 14, 19, 37, 70, 128, 195, 382, 671, 1188, 2143, 3550, 6519, 11544, 20263, 35316, 62302, 108272, 191277, 336749, 583438, 1022109, 1784180, 3115162, 5411730, 9454962, 16420712, 28558546, 49616719, 86004273, 149312549
Offset: 1
G.f.: A(x) is the limit of the composition of functions (x+x^n):
F_3(x) = x o x+x^2 o x+x^3 = x + x^2 + x^3 + 2*x^4 + x^6;
F_4(x) = F_3(x+x^4) = x + x^2 + x^3 + 3*x^4 + 2*x^5 + 4*x^6 +...
F_5(x) = F_4(x+x^5) = x + x^2 + x^3 + 3*x^4 + 3*x^5 + 6*x^6 +...
F_6(x) = F_5(x+x^6) = x + x^2 + x^3 + 3*x^4 + 3*x^5 + 7*x^6 +...
F_7(x) = x o x+x^2 o x+x^3 o x+x^4 o x+x^5 o x+x^6 o x+x^7 =
x + x^2 + x^3 + 3*x^4 + 3*x^5 + 7*x^6 + 14*x^7 + 18*x^8 +...
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{a(n)=local(F=x);if(n<1,0,for(k=2,n, F=subst(F,x,x+x^k+x*O(x^n)););return(polcoeff(F,n)))}
for(n=1,60,print1(a(n),", "))
A119471
G.f. A(x) equals the limit of the composition of functions (x+x^n) in reverse order; let F_1(x) = x, F_{n+1}(x) = F_n(x) + F_n(x)^(n+1), then A(x) = limit F_n(x): A(x) = ...o x+x^n o ... o x+x^3 o x+x^2 o x.
Original entry on oeis.org
1, 1, 1, 4, 8, 17, 50, 146, 399, 1087, 3042, 8741, 25509, 75259, 223529, 665215, 1983226, 5931158, 17800505, 53627756, 162206221, 492399027, 1499501067, 4579193127, 14017819056, 43001141630, 132154209754, 406818719006
Offset: 1
G.f.: A(x) is the limit of the composition of functions (x+x^n):
F_3(x) = x+x^3 o x+x^2 o x = x + x^2 + x^3 + 3*x^4 + 3*x^5 + x^6;
F_4(x) = x+x^4 o F_3(x) = x + x^2 + x^3 + 4*x^4 + 7*x^5 + 11*x^6 +...
F_5(x) = x+x^5 o F_4(x) = x + x^2 + x^3 + 4*x^4 + 8*x^5 + 16*x^6 +...
F_6(x) = x+x^6 o F_5(x) = x + x^2 + x^3 + 4*x^4 + 8*x^5 + 17*x^6 +...
F_7(x) = x+x^7 o x+x^6 o x+x^5 o x+x^4 o x+x^3 o x+x^2 o x =
x + x^2 + x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 50*x^7 + 145*x^8 +...
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{a(n)=local(F=x);if(n<1,0,for(k=2,n, F=subst(x+x^k,x,F+x*O(x^n)););return(polcoeff(F,n)))}
A163133
G.f. A(x) equals the composition of functions x*(1 + a(n)*x^n); let F_1(x) = x, F_{n+1}(x) = x*(1 + a(n)*F_n(x)^n), then A(x) = limit F_n(x): A(x) = ...o x*(1+a(n)*x^n) o...o x*(1+a(2)*x^2) o x*(1+a(1)*x).
Original entry on oeis.org
1, 1, 1, 4, 11, 42, 196, 988, 5314, 30724, 189962, 1244264, 8583042, 62112639, 469917193, 3707053139, 30421506639, 259182853643, 2288352779166, 20904648990227, 197305045507448, 1921509079008868, 19285642292478101
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + 4*x^4 + 11*x^5 + 42*x^6 + 196*x^7 +...
A(x) = ...o x*(1+11*x^5) o x*(1+4*x^4) o x*(1+x^3) o x*(1+x^2) o x*(1+x).
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{a(n)=local(F=x); if(n<1, 0, for(k=2, n, F=subst(x+a(k-1)*x^k +x*O(x^n),x,F); ); return(polcoeff(F, n)))}
Showing 1-3 of 3 results.