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.

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.

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

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 A277180.

Examples

			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! +...

Links

Crossrefs

Cf. A277183 (log(A(x)/x)), A277180, A136751.
Cf. A278332.

Programs

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

Formula

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,...

A277182 E.g.f.: logarithm of G(x)/x where G(x) = ... x*exp(x^4) o x*exp(x^3) o x*exp(x^2) o x*exp(x), a composition of functions x*exp(x^n) for n = 1,2,3,...

Original entry on oeis.org

1, 2, 18, 144, 1660, 27480, 548394, 12402992, 316789848, 9158652720, 296955697390, 10666960742328, 420121365404052, 17973670280757464, 828915057583647090, 40974375613614916320, 2161181874390019883056, 121176506157487442355168, 7199219738147125437960534, 451879288812982893026999720, 29885088906978769868636730540
Offset: 1

Views

Author

Paul D. Hanna, Oct 06 2016

Keywords

Comments

E.g.f. equals the logarithm of G(x)/x, where G(x) equals the e.g.f. of A277180.

Examples

			E.g.f.: A(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! + 420121365404052*x^13/13! + 17973670280757464*x^14/14! + 828915057583647090*x^15/15! + 40974375613614916320*x^16/16! +...
such that x*exp(A(x)) equals the infinite composition of functions:
x*exp(A(x)) = ... o x*exp(x^4) o x*exp(x^3) o x*exp(x^2) o x*exp(x),
which expands to begin:
x*exp(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! +...+ A277180(n)*x^n/n! +...
A related series expansion begins
exp(A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 241*x^4/4! + 3001*x^5/5! + 50011*x^6/6! + 1011067*x^7/7! + 23485729*x^8/8! + 617623489*x^9/9! + 18263529811*x^10/10!  + 601598367811*x^11/11! + 21859800985969*x^12/12! +...
GENERATING METHOD.
Once can generate the e.g.f. by the following process.
Start with L_1 = x, then continue
L_2 = L_1 + x^2*exp(2*L_1)
L_3 = L_2 + x^3*exp(3*L_2)
L_4 = L_3 + x^4*exp(4*L_3)
...
L_{n+1} = L_{n} + x^(n+1)*exp( (n+1)*L_{n} )
...
which tends to e.g.f. A(x) as a limit.
Explicitly, the initial functions are:
L_1 = x
L_2 = x + x^2*exp(2*x)
L_3 = x + x^2*exp(2*x) + x^3*exp(3*x +  3*x^2*exp(2*x) )
L_4 = x + x^2*exp(2*x) + x^3*exp(3*x +  3*x^2*exp(2*x) ) + x^4*exp(4*x +  4*x^2*exp(2*x) + 4*x^3*exp(3*x + 3*x^2*exp(2*x) ) )
L_5 = x + x^2*exp(2*x) + x^3*exp( 3*x + 3*x^2*exp(2*x) ) + x^4*exp( 4*x + 4*x^2*exp(2*x) + 4*x^3*exp( 3*x + 3*x^2*exp(2*x) ) ) + x^5*exp( 5*x + 5*x^2*exp(2*x) + 5*x^3*exp( 3*x + 3*x^2*exp(2*x) ) + 5*x^4*exp( 4*x + 4*x^2*exp(2*x) + 4*x^3*exp( 3*x + 3*x^2*exp(2*x) ) ) )
...
The derivative of these series may be computed like so
L_1' = 1
L_2' = L_1' + 2*x^1*exp( 2*L_1 ) * (1 + x*L_1')
L_3' = L_2' + 3*x^2*exp( 3*L_2 ) * (1 + x*L_2')
L_4' = L_3' + 4*x^3*exp( 4*L_3 ) * (1 + x*L_3')
...

Links

Crossrefs

Cf. A277180 (x*exp(A(x))), A277183.

Programs

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

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).
Showing 1-3 of 3 results.

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