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A144681 E.g.f. satisfies: A(x/A(x)) = exp(x). 4
1, 1, 3, 22, 305, 6656, 204337, 8226436, 414585425, 25315924960, 1828704716801, 153433983789164, 14739472821255481, 1602471473448455104, 195300935112810494801, 26470100501608768436716 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..136

FORMULA

E.g.f. satisfies: A(x) = exp( x*A(log A(x)) ).

E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(n+1).

E.g.f: A(x) = G(2x)^(1/2) where G(x/G(x)^2) = exp(x) and G(x) is the e.g.f. of A144682.

E.g.f: A(x) = G(3x)^(1/3) where G(x/G(x)^3) = exp(x) and G(x) is the e.g.f. of A144683.

E.g.f: A(x) = G(4x)^(1/4) where G(x/G(x)^4) = exp(x) and G(x) is the e.g.f. of A144684.

E.g.f: A(x) = 1/G(-x) where G(x*G(x)) = exp(x) and G(x) is the e.g.f. of A087961.

E.g.f. A(log(A(x))) = log(A(x))/x = G(x) is the e.g.f of A140049 where G(x) satisfies G(x*exp(-x*G(x))) = exp(x*G(x)).

EXAMPLE

E.g.f. A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 305*x^4/4! +...

A(x/A(x)) = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! +...

1/A(x) = 1 + x - x^2/2! + 10*x^3/3! - 159*x^4/4! + 3816*x^5/5! -+...

A(log(A(x)) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1005*x^4/4! + 26601*x^5/5! +...

ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(n+1) as follows.

Form a table of coefficients of x^k/k! in exp(x)*A(x)^n for n>=1, k>=0:

exp(x)*A(x)^1: [(1), 2, 6, 35, 416, 8437, 249340, ...];

exp(x)*A(x)^2: [1,(3), 13, 93, 1145, 22593, 645741, ...];

exp(x)*A(x)^3: [1, 4,(22), 181, 2320, 45199, 1257364, ...];

exp(x)*A(x)^4: [1, 5, 33,(305), 4097, 79825, 2177329, ...];

exp(x)*A(x)^5: [1, 6, 46, 471,(6656), 131001, 3529836, ...];

exp(x)*A(x)^6: [1, 7, 61, 685, 10201,(204337), 5477005, ...];

exp(x)*A(x)^7: [1, 8, 78, 953, 14960, 306643,(8226436), ...]; ...

then the terms along the main diagonal form this sequence shift left.

PROG

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(n=0, n, A=exp(serreverse(x/A))); n!*polcoeff(A, n)}

(PARI) {a(n)=local(A=1+x+sum(k=2, n-1, a(k)*x^k/k!)+x*O(x^n)); if(n==0, 1, (n-1)!*polcoeff(exp(x+x*O(x^n))*A^n, n-1))}

CROSSREFS

Cf. A144682, A144683, A144684, A087961, A140049.

Sequence in context: A326430 A119390 A271848 * A305963 A261280 A124567

Adjacent sequences:  A144678 A144679 A144680 * A144682 A144683 A144684

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 19 2008

STATUS

approved

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Last modified October 20 22:22 EDT 2021. Contains 348119 sequences. (Running on oeis4.)