A195947
E.g.f. satisfies: A(x) = Sum_{n>=0} (-1)^n/n! * Sum_{k=0..n} (-1)^k*C(n,k)*(1 + x*A(x)^k)^k.
Original entry on oeis.org
1, 1, 5, 58, 1093, 28731, 971719, 40236449, 1972617385, 111779567596, 7189852342091, 517600784497237, 41237095369088029, 3602389000897583001, 342422738142493542031, 35186740743134660359186, 3887047020291801938191057, 459397561144034558519708403
Offset: 0
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1093*x^4/4! + 28731*x^5/5! +...
where:
A(x) = 1 + A(x)*exp(A(x) - 1)*x + A(x)^4*exp(A(x)^2 - 1)*x^2/2! + A(x)^9*exp(A(x)^3 - 1)*x^3/3! + A(x)^16*exp(A(x)^4 - 1)*x^4/4! +...
Also, e.g.f. A = A(x) satisfies:
A(x) = 1 - (1 - (1+x*A)) + 1/2!*(1 - 2*(1+x*A) + (1+x*A^2)^2) -
1/3!*(1 - 3*(1+x*A) + 3*(1+x*A^2)^2 - (1+x*A^3)^3) +
1/4!*(1 - 4*(1+x*A) + 6*(1+x*A^2)^2 - 4*(1+x*A^3)^3 + (1+x*A^4)^4) -
1/5!*(1 - 5*(1+x*A) + 10*(1+x*A^2)^2 - 10*(1+x*A^3)^3 + 5*(1+x*A^4)^4 - (1+x*A^5)^5) +-...
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{a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n, A=1+sum(m=1,n,exp(A^m-1)*A^(m^2)*X^m/m!)); n!*polcoeff(A, n)}
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{a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n, A=1+sum(m=1, n, 1/m!*sum(k=0, m, binomial(m, k)*(-1)^(m-k)*(1+X*A^k)^k))); n!*polcoeff(A, n)}
A196125
E.g.f. satisfies: A(x) = Sum_{n>=0} exp(x*A(x)^(2*n) - 1)/n!.
Original entry on oeis.org
1, 1, 5, 61, 1241, 35321, 1293565, 57977333, 3074915569, 188387127793, 13093624617781, 1018008272854189, 87546809210311753, 8251472362124225001, 845856237057331033773, 93695948684292621079141, 11153023640354668717741793, 1419786005695986152133964769
Offset: 0
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 61*x^3/3! + 1241*x^4/4! + 35321*x^5/5! +...
where
A(x) = exp(x-1) + exp(x*A(x)^2-1) + exp(x*A(x)^4-1)/2! + exp(x*A(x)^6-1)/3! +...
Also,
A(x) = 1 + exp(A(x)^2-1)*x + exp(A(x)^4-1)*x^2/2! + exp(A(x)^6-1)*x^3/3! +...
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{a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n, A=1+sum(m=1, n, exp(A^(2*m)-1)*X^m/m!)); n!*polcoeff(A, n)}
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{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, 2*n+10, exp(x*A^(2*m)-1+x*O(x^n))/m!)); round(n!*polcoeff(A, n))}
A196531
E.g.f. satisfies: A(x) = Sum_{n>=0} exp(A(x)^n - 1)*A(x)^n*x^n/n!.
Original entry on oeis.org
1, 1, 5, 52, 853, 19291, 557719, 19657667, 817847321, 39236975668, 2132767035851, 129548751144077, 8696909403827077, 639445475893738749, 51105213341360790655, 4411321463887034379616, 409004744372281965629617, 40539225718259037965353203
Offset: 0
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 52*x^3/3! + 853*x^4/4! + 19291*x^5/5! +...
where
A(x) = 1 + exp(A(x)-1)*A(x)*x + exp(A(x)^2-1)*A(x)^2*x^2/2! + exp(A(x)^3-1)*A(x)^3*x^3/3! +...
Also, A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = e.g.f. of A195895(n):
G(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 201*x^4/4! + 2996*x^5/5! + 57613*x^6/6! +...+ A195895(n)*x^n/n! +...
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{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, exp(A^m-1+x*O(x^n))*A^m*x^m/m!)); n!*polcoeff(A, n)}
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/* Alternate e.g.f. (requires high precision): */
{a(n)=local(A=1+x); for(i=1, n, A=exp(-1)*sum(m=0, 2*n+10, exp(x*A^(m+1)+x*O(x^n))/m!)); round(n!*polcoeff(A, n))}
Showing 1-3 of 3 results.