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),", "))
A136751
G.f.: A(x) = ...o x/(1-x^4) o x/(1-x^3) o x/(1-x^2) o x/(1-x), composition of functions x/(1-x^n) for n = 1,2,3,...
Original entry on oeis.org
1, 1, 2, 5, 13, 36, 104, 310, 943, 2913, 9112, 28805, 91893, 295484, 956671, 3115805, 10200445, 33544983, 110755143, 366976365, 1219814018, 4066305982, 13590864072, 45534416250, 152895704998, 514446539489, 1734239511881
Offset: 0
G.f.: A(x) is the limit of composition of functions x/(1-x^n):
F_1(x) = x/(1-x)
F_2(x) = x/(1-x^2) o F_1(x) = x + x^2 + 2x^3 + 4x^4 + 8x^5 + 16x^6 +...
F_3(x) = x/(1-x^3) o F_2(x) = x + x^2 + 2x^3 + 5x^4 + 12x^5 + 30x^6 +...
F_4(x) = x/(1-x^4) o F_3(x) = x + x^2 + 2x^3 + 5x^4 + 13x^5 + 35x^6 +...
F_5(x) = x/(1-x^5) o F_4(x) = x + x^2 + 2x^3 + 5x^4 + 13x^5 + 36x^6 +...
F_6(x) = x/(1-x^6) o x/(1-x^5) o x/(1-x^4) o x/(1-x^3) o x/(1-x^2) o x/(1-x) =
x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 36*x^6 + 104*x^7 + 309*x^8 + 934*x^9 + ...
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{a(n)=local(A=x+x*O(x^n));if(n<=0,0,for(i=1,n,A=A/(1-A^i));polcoeff(A,n))}
for(n=1,40,print1(a(n),", "))
A136750
G.f.: A(x) = x/(1-x) o x/(1-x^2) o x/(1-x^3) o x/(1-x^4) o..., composition of functions x/(1-x^n) for n = ...,3,2,1.
Original entry on oeis.org
1, 1, 2, 4, 8, 17, 35, 74, 155, 329, 692, 1464, 3098, 6553, 13873, 29361, 62165, 131638, 278787, 590357, 1250337, 2648108, 5608735, 11879607, 25161906, 53295451, 112886028, 239108343, 506466366, 1072776347, 2272316123, 4813150684
Offset: 0
G.f.: A(x) is the limit of composition of functions x/(1-x^n):
F_1(x) = x/(1-x)
F_2(x) = F_1(x/(1-x^2)) = x + x^2 + 2x^3 + 3x^4 + 5x^5 + 8*x^6 + 13x^7 +...
F_3(x) = F_2(x/(1-x^3)) = x + x^2 + 2x^3 + 4x^4 + 7x^5 + 14x^6 + 26x^7 +...
F_4(x) = F_3(x/(1-x^4)) = x + x^2 + 2x^3 + 4x^4 + 8x^5 + 16x^6 + 32x^7 +...
F_5(x) = F_4(x/(1-x^5)) = x + x^2 + 2x^3 + 4x^4 + 8x^5 + 17x^6 + 34x^7 +...
F_6(x) = x/(1-x) o x/(1-x^2) o x/(1-x^3) o x/(1-x^4) o x/(1-x^5) o x/(1-x^6) =
x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 35*x^7 + 73*x^8 + 152*x^9 +...
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{a(n)=local(A=x+x*O(x^n));if(n<=0,0,for(i=1,n,A=A/(1-A^(n-i+1)));polcoeff(A,n))}
for(n=1,30,print1(a(n),", "))
A119472
G.f. A(x) equals the composition of functions x*(1 + a(n)*x^n); let F_1(x) = x, F_{n+1}(x) = F_n( x*(1 + a(n)*x^n) ), then A(x) = limit F_n(x): A(x) = x*(1+a(1)*x) o x*(1+a(2)*x^2) o ... o x*(1+a(n)*x^n) o ...
Original entry on oeis.org
1, 1, 1, 3, 5, 15, 42, 124, 352, 1124, 3574, 11588, 38033, 127297, 426302, 1459632, 4986161, 17345028, 60373874, 212488958, 747271311, 2661073611, 9451241495, 33925353554, 121618969926, 439680022154, 1586931378911, 5775629048634
Offset: 1
G.f.: A(x) is the limit of the composition of x*(1+a(n)*x^n):
F_3(x) = 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+3*x^5) = x + x^2 + x^3 + 3*x^4 + 5*x^5 + 10*x^6 +...
F_6(x) = F_5(x+5*x^6) = x + x^2 + x^3 + 3*x^4 + 5*x^5 + 15*x^6 +...
F_7(x) = x+1*x^2 o x+1*x^3 o x+1*x^4 o x+3*x^5 o x+5*x^6 o x+15*x^7 =
x + x^2 + x^3 + 3*x^4 + 5*x^5 + 15*x^6 + 42*x^7 + 82*x^8 +...
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{a(n)=local(F=x);if(n<1,0,for(k=2,n, F=subst(F,x,x+a(k-1)*x^k +x*O(x^n)););return(polcoeff(F,n)))}
A136752
G.f.: A(x) = x/(1-x) o x/(1-x^2) o x/(1-x^4) o x/(1-x^8) o..., composition of functions x/(1 - x^{2^n}) for n=0,1,2,3,...
Original entry on oeis.org
1, 1, 2, 3, 6, 10, 19, 33, 61, 108, 198, 354, 645, 1159, 2106, 3795, 6874, 12405, 22457, 40560, 73374, 132578, 239782, 433362, 783602, 1416401, 2560953, 4629393, 8369741, 15130440, 27354520, 49451349, 89401972, 161622356, 292191262
Offset: 0
G.f.: A(x) is the limit of composition of functions x/(1-x^{2^n}):
F_0(x) = x/(1-x)
F_1(x) = F_1(x/(1-x^2)) = x + x^2 + 2x^3 + 3x^4 + 5x^5 + 8*x^6 + 13x^7 +...
F_2(x) = F_2(x/(1-x^4)) = x + x^2 + 2x^3 + 3x^4 + 6x^5 + 10x^6 + 19x^7 +...
F_3(x) = x/(1-x) o x/(1-x^2) o x/(1-x^4) o x/(1-x^8) =
x + x^2 + 2x^3 + 3x^4 + 6x^5 + 10x^6 + 19x^7 + 33x^8 + 61x^9 + 108x^10 +...
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{a(n)=local(A=x+x*O(x^n));if(n<=0,0,m=#binary(n+1); for(i=1,m,A=A/(1-A^(2^(m-i))));polcoeff(A,n))}
A136753
G.f.: A(x) = ...o x/(1-x^8) o x/(1-x^4) o x/(1-x^2) o x/(1-x), composition of functions x/(1 - x^{2^n}) for n=...,3,2,1,0.
Original entry on oeis.org
1, 1, 2, 4, 9, 21, 52, 134, 355, 955, 2590, 7052, 19246, 52638, 144368, 397468, 1099720, 3060936, 8577496, 24210808, 68843806, 197176726, 568585576, 1649739332, 4812731846, 14105205846, 41498665884, 122469937048
Offset: 1
G.f.: A(x) is the limit of composition of functions x/(1-x^{2^n}):
F_0(x) = x/(1-x)
F_1(x) = x/(1-x^2) o F_0(x) = x + x^2 + 2x^3 + 4x^4 + 8x^5 + 16x^6 +...
F_2(x) = x/(1-x^4) o F_1(x) = x + x^2 + 2x^3 + 4x^4 + 9x^5 + 21x^6 +...
F_3(x) = x/(1-x^8) o F_2(x) = x + x^2 + 2x^3 + 4x^4 + 9x^5 + 21x^6 +...
F_4(x) = x/(1-x^16) o x/(1-x^8) o x/(1-x^4) o x/(1-x^2) o x/(1-x) =
x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 21*x^6 + 52*x^7 + 134*x^8 + 355*x^9 +...
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{a(n)=local(A=x+x*O(x^n));if(n<=0,0,for(i=0,#binary(n+1),A=A/(1-A^(2^i)));polcoeff(A,n))}
Showing 1-6 of 6 results.
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