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A302103 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (2 + x*A(x)^n)^n / 3^(n+1). 4
1, 1, 6, 63, 837, 12672, 208686, 3647568, 66697203, 1264307667, 24696153573, 495076265421, 10157438738790, 212900154037875, 4553735135491134, 99341289091151409, 2210262851488661562, 50173932628981325523, 1162965513498859292415, 27554435907912281877315, 668277970101220006626558, 16617278354076763108026795 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare to: G(x) = Sum_{n>=0} (2 + x*G(x)^k)^n / 3^(n+1) holds when G(x) = 1 + x*G(x)^(k+1) for fixed k.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..100

FORMULA

G.f. A(x) satisfies:

(1) A(x) = Sum_{n>=0} (2 + x*A(x)^n)^n / 3^(n+1).

(2) A(x) = Sum_{n>=0} x^n * A(x)^(n^2) / (3 - 2*A(x)^n)^(n+1).

EXAMPLE

G.f.: A(x) = 1 + x + 6*x^2 + 63*x^3 + 837*x^4 + 12672*x^5 + 208686*x^6 + 3647568*x^7 + 66697203*x^8 + 1264307667*x^9 + 24696153573*x^10 + ...

such that

A(x) = 2/3 + (2 + x*A(x))/3^2 + (2 + x*A(x)^2)^2/3^3 + (2 + x*A(x)^3)^3/3^4 + (2 + x*A(x)^4)^4/3^5 + (2 + x*A(x)^5)^5/3^6 + (2 + x*A(x)^6)^6/3^7 + ...

Also, due to a series identity,

A(x) = 1 + x*A(x)/(3 - 2*A(x))^2 + x^2*A(x)^4/(3 - 2*A(x)^2)^3 + x^3*A(x)^9/(3 - 2*A(x)^3)^4 + x^4*A(x)^16/(3 - 2*A(x)^4)^5 + x^5*A(x)^25/(3 - 2*A(x)^5)^6 + x^6*A(x)^36/(3 - 2*A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (3 - 2*A(x)^n)^(n+1) + ...

PROG

(PARI) {a(n) = my(A=1); for(i=0, n, A = sum(m=0, n, x^m * A^(m^2) / (3 - 2*A^m + x*O(x^n))^(m+1) )); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A300050, A302104, A302105.

Sequence in context: A341376 A234465 A231552 * A229451 A132078 A113669

Adjacent sequences:  A302100 A302101 A302102 * A302104 A302105 A302106

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 05 2018

STATUS

approved

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Last modified August 1 13:55 EDT 2021. Contains 346391 sequences. (Running on oeis4.)