A136751 -id:A136751 - cp's OEIS frontend

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.

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

Paul D. Hanna, Jan 21 2008

nonn

The composition transpose of A136751.

			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 +...

Cf. A136751; variants: A136752, A136753, A119470, A119471.

{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),", "))

A277180 E.g.f.: A(x) = ... xexp(x^4) o xexp(x^3) o xexp(x^2) o xexp(x), the composition of functions x*exp(x^n) for n = 1,2,3,...

Original entry on oeis.org

1, 2, 9, 100, 1205, 18006, 350077, 8088536, 211371561, 6176234890, 200898827921, 7219180413732, 284177412817597, 12162803253287246, 562046000617917285, 27867599169654763696, 1475047571057004959057, 83000104748219010488850, 4947512767013757600177049, 311464596400042198210554620, 20652342444419128752639269541, 1438800618216725748602640496342

Offset: 1

Paul D. Hanna, Oct 04 2016

nonn

The compositional transpose of functions x*exp(x^n) yields the e.g.f. of A277181.

			E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 100*x^4/4! + 1205*x^5/5! + 18006*x^6/6! + 350077*x^7/7! + 8088536*x^8/8! + 211371561*x^9/9! + 6176234890*x^10/10! + 200898827921*x^11/11! + 7219180413732*x^12/12! +...
such that A(x) is the limit of composition of functions x*exp(x^n):
A(x) = ... o x*exp(x^5) o x*exp(x^4) o x*exp(x^3) o x*exp(x^2) o x*exp(x)
working from right to left.
Illustration of generating method.
Start with F_0(x) = x and then continue as follows.
F_1(x) = x*exp(x),
F_2(x) = F_1(x) * exp( F_1(x)^2 ),
F_3(x) = F_2(x) * exp( F_2(x)^3 ),
F_4(x) = F_3(x) * exp( F_3(x)^4 ),
...
F_{n+1}(x) = F_{n}(x) * exp( F_{n}(x)^(n+1) )
...
the limit of which equals the e.g.f. A(x).
The above series begin:
F_1(x) = x + 2*x^2/2! + 3*x^3/3! + 4*x^4/4! + 5*x^5/5! + 6*x^6/6! +...
F_2(x) = x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 605*x^5/5! + 5046*x^6/6! +...
F_3(x) = x + 2*x^2/2! + 9*x^3/3! + 100*x^4/4! + 1085*x^5/5! + 13686*x^6/6! +...
F_4(x) = x + 2*x^2/2! + 9*x^3/3! + 100*x^4/4! + 1205*x^5/5! + 17286*x^6/6! +...
...
RELATED SERIES.
The logarithm of A(x)/x begins:
log(A(x)/x) = x + 2*x^2/2! + 18*x^3/3! + 144*x^4/4! + 1660*x^5/5! + 27480*x^6/6! + 548394*x^7/7! + 12402992*x^8/8! + 316789848*x^9/9! + 9158652720*x^10/10! + 296955697390*x^11/11! + 10666960742328*x^12/12! +...+ A277182(n)*x^n/n! +...
The series reversion of the e.g.f. begins:
Series_Reversion(A(x)) = x - 2*x^2/2! + 3*x^3/3! - 40*x^4/4! + 505*x^5/5! - 4776*x^6/6! + 53179*x^7/7! - 1065296*x^8/8! + 25478289*x^9/9! - 480072880*x^10/10! + 9400182451*x^11/11! - 300620572968*x^12/12! +...

Vaclav Kotesovec, Table of n, a(n) for n = 1..300

Cf. A277182 (log A(x)/x), A277181, A136751.

Cf. A278332.

{a(n) = my(A=x +x*O(x^n)); if(n<=0, 0, for(i=1, n, A = A*exp(A^i)); n!*polcoeff(A, n))}
for(n=1,30,print1(a(n),", "))

{a(n) = my(A=x +x*O(x^n)); if(n<=0, 0, for(i=1, n, A = subst(A,x, x*exp(x^(n-i+1) +x*O(x^n))))); n!*polcoeff(A, n)}
for(n=1,30,print1(a(n),", "))

E.g.f. A(x) satisfies: Series_Reversion(A(x)) = LambertW(x) o (LambertW(2*x^2)/2)^(1/2) o (LambertW(3*x^3)/3)^(1/3) o (LambertW(4*x^4)/4)^(1/4) o ..., the composition of functions (LambertW(n*x^n)/n)^(1/n) for n = ...,3,2,1.

A277181 E.g.f.: A(x) = xexp(x) o xexp(x^2) o xexp(x^3) o xexp(x^4) o ..., the composition of functions x*exp(x^n) for n=...,3,2,1.

Original entry on oeis.org

1, 2, 9, 76, 605, 7326, 97237, 1414904, 24130521, 467773210, 9636459041, 215484787332, 5351427245749, 141098897750006, 3995090542811565, 120415709525270896, 3833710980240095537, 130061101059127375794, 4649348119132468282681, 174231442774945244111420, 6859230825811289134828941, 282654139723294546295799502, 12162998707984268597918477189, 546138551651775603897277518696

Offset: 1

Paul D. Hanna, Oct 04 2016

nonn

The compositional transpose of functions x*exp(x^n) yields the e.g.f. of A277180.

			E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 605*x^5/5! + 7326*x^6/6! + 97237*x^7/7! + 1414904*x^8/8! + 24130521*x^9/9! + 467773210*x^10/10! + 9636459041*x^11/11! + 215484787332*x^12/12! +...
such that A(x) is the limit of composition of functions x*exp(x^n):
A(x) = x*exp(x) o x*exp(x^2) o x*exp(x^3) o x*exp(x^4) o x*exp(x^5) o ...
working from left to right.
Illustration of generating method.
Start with F_0(x) = x and then continue as follows.
F_1(x) = x*exp(x),
F_2(x) = F_1( x*exp(x^2) ),
F_3(x) = F_2( x*exp(x^3) ),
F_4(x) = F_3( x*exp(x^4) ),
F_5(x) = F_4( x*exp(x^5) ),
...
F_{n+1}(x) = F_{n}( x*exp(x^(n+1)) ),
...
the limit of which equals the e.g.f. A(x).
The above series begin:
F_1(x) = x + 2*x^2/2! + 3*x^3/3! + 4*x^4/4! + 5*x^5/5! + 6*x^6/6! +...
F_2(x) = x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 245*x^5/5! + 1926*x^6/6! +...
F_3(x) = x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 485*x^5/5! + 5166*x^6/6! +...
F_4(x) = x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 605*x^5/5! + 6606*x^6/6! +...
F_5(x) = x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 605*x^5/5! + 7326*x^6/6! +...
...
A related series begins:
Series_Reversion(A(x)) = x - 2*x^2/2! + 3*x^3/3! - 16*x^4/4! + 385*x^5/5! - 6696*x^6/6! + 104419*x^7/7! - 1785344*x^8/8! + 37367649*x^9/9! - 986989600*x^10/10! + 30811625251*x^11/11! - 1031073660288*x^12/12! +...

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A277183 (log(A(x)/x)), A277180, A136751.

Cf. A278332.

{a(n) = my(A=x +x*O(x^n)); if(n<=0, 0, for(i=1, n, A = subst(A,x, x*exp(x^i +x*O(x^n))))); n!*polcoeff(A, n)}
for(n=1,30,print1(a(n),", "))

{a(n) = my(A=x+x*O(x^n)); if(n<=0, 0, for(i=1, n, A = A*exp(A^(n-i+1)))); n!*polcoeff(A, n)}
for(n=1,30,print1(a(n),", "))

E.g.f. A(x) satisfies: Series_Reversion(A(x)) = ... (LambertW(4*x^4)/4)^(1/4) o (LambertW(3*x^3)/3)^(1/3) o (LambertW(2*x^2)/2)^(1/2) o LambertW(x), the composition of functions (LambertW(n*x^n)/n)^(1/n) for n = 1,2,3,...

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

Paul D. Hanna, Jan 21 2008

nonn

The composition transpose of A136753.

			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 +...

Cf. A136753; variants: A136750, A136751, A119470, A119471.

{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

Paul D. Hanna, Jan 21 2008

nonn

The composition transpose of A136752.

			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 +...

Cf. A136752; variants: A136750, A136751, A119470, A119471.

{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))}

A206721 G.f.: A(x) = ...o x/(1-x^7) o x/(1-x^5) o x/(1-x^3) o x/(1-x), a composition of functions x/(1-x^(2*n-1)) for n=...3,2,1.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 28, 65, 155, 387, 1002, 2641, 7016, 18743, 50321, 135608, 366399, 992221, 2693705, 7333666, 20026582, 54858122, 150739957, 415479332, 1148563576, 3183963911, 8848963329, 24650364462, 68810859415, 192439812663, 539072534278, 1512285566266

Offset: 1

Paul D. Hanna, Feb 11 2012

nonn

Compositional transpose of A206720.

			G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 12*x^6 + 28*x^7 + 65*x^8 +...
where A(x) is the limit of composition of functions x/(1-x^(2*n-1)):
F_1(x) = x/(1-x)
F_2(x) = x/(1-x^3) o F_1(x) = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 11*x^6 +...
F_3(x) = x/(1-x^5) o F_2(x) = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 12*x^6 +...
F_4(x) = x/(1-x^7) o x/(1-x^5) o x/(1-x^3) o x/(1-x); ...

Cf. A206720, A136751.

{a(n)=local(A=x+x*O(x^n)); if(n<=0, 0, for(i=1, n, A=A/(1-A^(2*i-1))); polcoeff(A, n))}
for(n=1,45,print1(a(n),", "))

cp's OEIS Frontend

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.

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A277180 E.g.f.: A(x) = ... x*exp(x^4) o x*exp(x^3) o x*exp(x^2) o x*exp(x), the composition of functions x*exp(x^n) for n = 1,2,3,...

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A277181 E.g.f.: A(x) = x*exp(x) o x*exp(x^2) o x*exp(x^3) o x*exp(x^4) o ..., the composition of functions x*exp(x^n) for n=...,3,2,1.

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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,...

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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.

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A206721 G.f.: A(x) = ...o x/(1-x^7) o x/(1-x^5) o x/(1-x^3) o x/(1-x), a composition of functions x/(1-x^(2*n-1)) for n=...3,2,1.

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A277180 E.g.f.: A(x) = ... xexp(x^4) o xexp(x^3) o xexp(x^2) o xexp(x), the composition of functions x*exp(x^n) for n = 1,2,3,...

A277181 E.g.f.: A(x) = xexp(x) o xexp(x^2) o xexp(x^3) o xexp(x^4) o ..., the composition of functions x*exp(x^n) for n=...,3,2,1.