A144684 - OEIS

A144684

E.g.f. satisfies: A(x/A(x)^4) = exp(x).

1, 1, 9, 265, 15281, 1379441, 173762425, 28528990393, 5838405067745, 1443332192692321, 421171717332106601, 142520112730201819625, 55149333235223148407953, 24128837930726025305020369

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..13.

FORMULA

E.g.f. satisfies: A(x) = exp( x*A(log A(x))^4 ).

E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(4*n+4).

E.g.f. A(x) = G(4x)^(1/4) where G(x/G(x)) = exp(x) and G(x) is the g.f. of A144681.

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

EXAMPLE

E.g.f. A(x) = 1 + x + 9*x^2/2! + 265*x^3/3! + 15281*x^4/4! +...

ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(4*n+4) as follows.

Form a table of coefficients of x^k/k! in exp(x)*A(x)^(4*n) for n>=1, k>=0:

exp(x)*A(x)^4: [(1), 5, 57, 1565, 84017, 7220725, 879058921, ...];

exp(x)*A(x)^8: [1, (9), 145, 4377, 231713, 19222569, 2265114033, ...];

exp(x)*A(x)^12:[1, 13,(265), 8821, 472273, 38328733, 4391383897, ...];

exp(x)*A(x)^16:[1, 17, 417,(15281), 841025, 67630417, 7578669793, ...];

exp(x)*A(x)^20:[1, 21, 601, 24141, (1379441), 111109701, 12258211401, ...];

exp(x)*A(x)^24:[1, 25, 817, 35785, 2135137, (173762425), 18997491601, ...];

exp(x)*A(x)^28:[1, 29, 1065, 50597, 3161873, 261721069,(28528990393), ...]; ...

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^4))); 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^(4*n), n-1))}

CROSSREFS

Cf. A144681, A144682, A144683.

Sequence in context: A202884 A188963 A306535 * A191960 A168600 A088672

Adjacent sequences: A144681 A144682 A144683 * A144685 A144686 A144687

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 19 2008

STATUS

approved