A140055
E.g.f.: A(x) = G(G(x)) where G(x) = x*exp(A(x)) such that G( x*exp(-G(x)) ) = x and G(x) is the e.g.f. of A140054.
Original entry on oeis.org
1, 4, 42, 764, 20400, 731862, 33397168, 1867950856, 124680486816, 9733666171850, 874978919826264, 89437471672859532, 10289414670501314608, 1320997962702267801070, 187894667581541881127640, 29426125555003596239544848, 5046809953516305090792395328
Offset: 1
E.g.f.: A(x) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...
x*exp(A(x)) = x + 2*x^2/2! + 15*x^3/3! + 220*x^4/4! + 5025*x^5/5! +...
where G(x) = x*exp(A(x)) satisfies G(G(x)) = A(x).
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b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*
b(j-1, j)*j*b(n-j, k)*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n$2):
seq(a(n), n=1..20); # Alois P. Heinz, Aug 21 2019
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m = 20; G[_] = 0;
Do[G[x_] = x Exp[G[G[x]]] + O[x]^(m+1) // Normal, {m}];
CoefficientList[G[G[x]]/x + O[x]^m, x]*Range[m]! (* Jean-François Alcover, Oct 14 2019 *)
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{a(n)=local(A=x);for(i=0,n,A=x*exp(subst(A,x,A+x*O(x^n))));n!*polcoeff(subst(A,x,A),n)}
A140049
E.g.f. A(x) satisfies: A( x*exp(-x*A(x)) ) = exp(x*A(x)).
Original entry on oeis.org
1, 1, 5, 55, 1005, 26601, 941863, 42372177, 2336926665, 153927536545, 11869936146891, 1055015092106889, 106731589524249517, 12163935655214359329, 1548324822731892094191, 218516875165035215308801, 33979477899236956531288977, 5790103152487972170694748097
Offset: 0
A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1005*x^4/4! + 26601*x^5/5! +...
Log(A(x)) = G(x) = e.g.f. of A140055:
Log(A(x)) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...
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b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*j
*b(j-1, j)*b(n-j, k)*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n, n+1):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 21 2019
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m = 18; A[_] = 0;
Do[A[x_] = Exp[x A[x] A[x A[x]]] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] * Range[0, m-1]! (* Jean-François Alcover, Oct 03 2019 *)
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{a(n)=local(A=x);for(i=0,n,A=serreverse(x*exp(-A+x*O(x^n))));n!*polcoeff(A,n+1)}
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{a(n)=local(A=x);for(i=0,n,A=x*exp(subst(A,x,A+x*O(x^n))));n!*polcoeff(A,n+1)}
From Paul D. Hanna, Jul 09 2009: (Start)
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{a(n,m=1)=if(n==0,1,if(m==0,0^n,sum(k=0,n,binomial(n,k)*m*(n+m)^(k-1)*a(n-k,k))))}
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/* Log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n! where: */
{L(n)=if(n<1,0,sum(k=1,n,binomial(n,k)*n^(k-1)*a(n-k,k)))} (End)
A178533
E.g.f. satisfies: A(x) = Sum_{n>=0} A_{n}(x)^(n+1)/n! where A_{n}(x) denotes the n-th iteration of A(x).
Original entry on oeis.org
1, 2, 15, 220, 5025, 159066, 6515551, 330344736, 20099812977, 1434653999650, 118051188070431, 11045624947772328, 1162186160312303737, 136250833163724567666, 17661781851438961602255, 2514858304152906320391136
Offset: 1
E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 220*x^4/4! + 5025*x^5/5! +...
The e.g.f. equals the series:
A(x) = x + A(x)^2 + A(A(x))^3/2! + A(A(A(x)))^4/3! + A(A(A(A(x))))^5/4! +...+ A_{n}(x)^(n+1)/n! +...
Related expansions:
A(A(x)) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...
A(A(A(x))) = x + 6*x^2/2! + 81*x^3/3! + 1776*x^4/4! + 55125*x^5/5! +...
A(A(A(A(x)))) = x + 8*x^2/2! + 132*x^3/3! + 3400*x^4/4! + 121080*x^5/5! +...
The series reversion of the e.g.f. is given by:
A_{-1}(x) = x - x^2 - A(A(x))^3/2! - A(A(A(x)))^4/3! - A(A(A(A(x))))^5/4! -...- A_{n-1}(x)^(n+1)/n! -...
A_{-1}(x) = x - 2*x^2/2! - 3*x^3/3! - 40*x^4/4! - 795*x^5/5! - 22356*x^6/6! -...
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{ITERATE(F,n,p)=local(G=x);for(i=1,n,G=subst(F,x,G+x*O(x^p)));G}
{a(n)=local(A=x+x^2+x*O(x^n));for(i=1,n,A=sum(m=0,n,ITERATE(A,m,n)^(m+1)/m!));n!*polcoeff(A,n)}
Showing 1-3 of 3 results.
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