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A221096 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(2*n))^n/n!. 3
1, 1, 4, 42, 768, 19460, 637200, 25724916, 1233957312, 68591031120, 4338982958400, 307907317681920, 24229505587541760, 2094548798610726432, 197370092438311892736, 20140182770328963216000, 2213078753956025271214080, 260601290312643875434817280 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..17.

FORMULA

E.g.f. also satisfies:

(1) A(x) = Sum_{n>=0} binomial(A(x)^(2*n), n) * x^n.

(2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(x)^(2*n*k)/n!.

EXAMPLE

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 19460*x^5/5! +...

where A(x) satisfies:

A(x) = 1 + log(1 + x*A(x)^2) + log(1 + x*A(x)^4)^2/2! + log(1 + x*A(x)^6)^3/3! +...

The e.g.f. also satisfies:

A(x) = 1 + A(x)^2*x + A(x)^4*(A(x)^4-1)*x^2/2! + A(x)^6*(A(x)^6-1)*(A(x)^6-2)*x^3/3! + A(x)^8*(A(x)^8-1)*(A(x)^8-2)*(A(x)^8-3)*x^4/4! +...+ binomial(A(x)^(2*n), n)*x^n +...

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+x*(A+x*O(x^n))^(2*m))^m/m!)); n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, binomial((A+x*O(x^n))^(2*m), m)*x^m)); n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*(A+x*O(x^n))^(2*m*k))*x^m/m!)); n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A189981, A221097, A221098, A221099.

Sequence in context: A295763 A266526 A140055 * A179499 A268567 A197866

Adjacent sequences:  A221093 A221094 A221095 * A221097 A221098 A221099

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 01 2013

STATUS

approved

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Last modified September 26 14:21 EDT 2022. Contains 356999 sequences. (Running on oeis4.)