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.
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 181*x^4/4! + 2111*x^5/5! + 31471*x^6/6! + 563305*x^7/7! + 11634169*x^8/8! + 275850685*x^9/9! +... RELATED SERIES. log(A(x)) = x + 2*x^2/2! + 15*x^3/3! + 96*x^4/4! + 1100*x^5/5! + 15870*x^6/6! + 278817*x^7/7! + 5558616*x^8/8! + 129258684*x^9/9! +...
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! +...
{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) = 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! +...
{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),", "))
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