cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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,...

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

Views

Author

Paul D. Hanna, Oct 04 2016

Keywords

Comments

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

Examples

			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! +...

Links

Crossrefs

Cf. A277182 (log A(x)/x), A277181, A136751.
Cf. A278332.

Programs

  • PARI
    {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),", "))
  • PARI
    {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),", "))

Formula

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.

A278332 E.g.f. equals the limit of the average of all permutations of the compositions of the functions x*exp(x^k), for k=1..n, as n increases.

Original entry on oeis.org

1, 2, 9, 88, 905, 12666, 220297, 4506440, 104707521, 2758506850
Offset: 1

Views

Author

Paul D. Hanna, Nov 18 2016

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 88*x^4/4! + 905*x^5/5! + 12666*x^6/6! + 220297*x^7/7! + 4506440*x^8/8! + 104707521*x^9/9! + 2758506850*x^10/10! +...
Generating method.
Define F(n,x) as the average of the sum over all n! permutations of the compositions of x*exp(x^k) for k=1..n, then the e.g.f. of this sequence is the limit of the functions F(n,x) as n grows.
Examples of some initial functions F(n,x) are as follows.
At n=1, F(1,x) = x*exp(x).
At n=2, F(2,x) = (1/2!)*([x*exp(x) o x*exp(x^2)] + [x*exp(x^2) o x*exp(x)]) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 425*x^5/5! + 3486*x^6/6! +...
At n=3, F(3,x) = (1/3!)*([x*exp(x) o x*exp(x^2) o x*exp(x^3)] + [x*exp(x^2) o x*exp(x) o x*exp(x^3)] + [x*exp(x) o x*exp(x^3) o x*exp(x^2)] + [x*exp(x^3) o x*exp(x) o x*exp(x^2)] + [x*exp(x^2) o x*exp(x^3) o x*exp(x)] + [x*exp(x^3) o x*exp(x^2) o x*exp(x)]) = x + 2*x^2/2! + 9*x^3/3! + 88*x^4/4! + 785*x^5/5! + 9426*x^6/6! +...
etc.

Crossrefs

Cf. A278333(n) = a(n+1)/(n+1) for n>=0.
Cf. A277180, A277181.

Programs

  • PARI
    /* Informal code to generate terms (ran over 100 hrs for 10 terms!) */
{F(n) = x*exp(x^n +x*O(x^16))}
{P(n) = vector(n!,k, numtoperm(n,k) )}
{A(n) = my(H=0); for(k=1,n!, G=x; for(j=1,n, G=subst(G,x, F(P(n)[k][j]) )); H=H+G);H/n!}
{Vec(serlaplace(A(6)))}

Formula

E.g.f.: A(x) = limit_{n->oo} (1/n!) * Sum(all permutations of compositions of functions x*exp(x^k) for k=1..n).

A277183 E.g.f.: logarithm of G(x)/x where G(x) = x*exp(x) o x*exp(x^2) o x*exp(x^3) o x*exp(x^4) o ..., a composition of functions x*exp(x^n) for n = ...,3,2,1.

Original entry on oeis.org

1, 2, 12, 48, 540, 5040, 53760, 745920, 12897360, 206236800, 3682657440, 83366236800, 1849952744640, 45496897205760, 1176291493977600, 32097739145472000, 983522184476832000, 31186123650255974400, 1021800029809932172800, 35859334900163943168000, 1320567044275182140236800, 50849016060752080883712000, 2063131407115000462188748800, 86946484539570696351914803200
Offset: 1

Views

Author

Paul D. Hanna, Oct 07 2016

Keywords

Comments

E.g.f. equals the logarithm of G(x)/x, where G(x) equals the e.g.f. of A277181.

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 48*x^4/4! + 540*x^5/5! + 5040*x^6/6! + 53760*x^7/7! + 745920*x^8/8! + 12897360*x^9/9! + 206236800*x^10/10! + 3682657440*x^11/11! + 83366236800*x^12/12! + 1849952744640*x^13/13! + 45496897205760*x^14/14! + 1176291493977600*x^15/15! + 32097739145472000*x^16/16! +...
such that x*exp(A(x)) equals the infinite composition of functions:
x*exp(A(x)) =  x*exp(x) o x*exp(x^2) o x*exp(x^3) o x*exp(x^4) o ...,
which expands to begin:
x*exp(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! +...+ A277181(n)*x^n/n! +...
A related series expansion begins
exp(A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 121*x^4/4! + 1221*x^5/5! + 13891*x^6/6! + 176863*x^7/7! + 2681169*x^8/8! + 46777321*x^9/9! + 876041731*x^10/10! + 17957065611*x^11/11! + 411648249673*x^12/12! +...
GENERATING METHOD.
One can generate the e.g.f. by the following process.
Find A(x) = L_1 by working backward using the relation
L_1 = L_2 + x^1*exp(1*L_2)
L_2 = L_3 + x^2*exp(2*L_3)
L_3 = L_4 + x^3*exp(3*L_4)
L_4 = L_5 + x^4*exp(4*L_5)
...
L_{n} = L_{n+1} + x^n*exp( n*L_{n+1} )
...
Explicitly, the initial series begin:
L_1 = x + x^2 + 2*x^3 + 2*x^4 + 9/2*x^5 + 7*x^6 + 32/3*x^7 + 37/2*x^8 +...
L_2 = x^2 + x^3 + x^4 + 3*x^5 + 3*x^6 + 6*x^7 + 8*x^8 + 20*x^9 + 22*x^10 +...
L_3 = x^3 + x^4 + x^5 + x^6 + 4*x^7 + 4*x^8 + 8*x^9 + 8*x^10 + 35/2*x^11 +...
L_4 = x^4 + x^5 + x^6 + x^7 + x^8 + 5*x^9 + 5*x^10 + 10*x^11 + 10*x^12 +...
L_5 = x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + 6*x^11 + 6*x^12 + 12*x^13 + 12*x^14 +...
L_6 = x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + 7*x^13 + 7*x^14 + 14*x^15 + 14*x^16 +...
L_7 = x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + 8*x^15 + 8*x^16 + 16*x^17 + 16*x^18 +...
L_8 = x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + 9*x^17 + 9*x^18 + 18*x^19 + 18*x^20 +...
...

Crossrefs

Cf. A277181 (x*exp(A(x))), A277182.

Programs

  • PARI
    {a(n) = my(A=x +x^2*O(x^n)); if(n<=0, 0, for(i=1, n, A = subst(A, x, x*exp(x^i +x^2*O(x^n))))); n!*polcoeff(log(A/x), n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=x +x^2*O(x^n)); if(n<=0, 0, for(i=1, n, A = A*exp(A^(n-i+1)))); n!*polcoeff(log(A/x), n)}
for(n=1, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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