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A325574 G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x)^n * (A(x)^n + (-1)^n)^n. 1
1, 4, 32, 400, 6016, 99968, 1779456, 33343488, 650141696, 13084840960, 270257033216, 5704378748928, 122667151491072, 2681371746680832, 59480466149277696, 1337376871507230720, 30452122562013954048, 701788662787645112320, 16362889157705834954752, 385927625558617880526848, 9207728256045430003990528, 222272707783603972590272512, 5430861508857531736344494080, 134380300947604255424846495744 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) appears to be divisible by 2^n for n >= 0.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..200

FORMULA

G.f. A(x) satisfies:

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

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

EXAMPLE

G.f.: A(x) = 1 + 4*x + 32*x^2 + 400*x^3 + 6016*x^4 + 99968*x^5 + 1779456*x^6 + 33343488*x^7 + 650141696*x^8 + 13084840960*x^9 + 270257033216*x^10 + ...

such that

1 = 1 - x*(A(x) - 1) + x^2*(A(x)^2 + 1)^2 - x^3*(A(x)^3 - 1)^3 + x^4*(A(x)^4 + 1)^4 - x^5*(A(x)^5 - 1)^5 + x^6*(A(x)^6 + 1)^6 - x^7*(A(x)^7 - 1)^7 + x^8*(A(x)^8 + 1)^8 - x^9*(A(x)^9 - 1)^9 + x^10*(A(x)^10 + 1)^10 + ...

also,

1 = 1/(1 - x) - x*A(x)/(1 + x*A(x))^2 + x^2*A(x)^4/(1 - x*A(x)^2)^3 - x^3*A(x)^9/(1 + x*A(x)^3)^4 + x^4*A(x)^16/(1 - x*A(x)^4)^5 - x^5*A(x)^25/(1 + x*A(x)^5)^6 + x^6*A(x)^36/(1 - x*A(x)^6)^7 - x^7*A(x)^49/(1 + x*A(x)^7)^8 + x^8*A(x)^64/(1 - x*A(x)^8)^9 - x^9*A(x)^81/(1 + x*A(x)^9)^10 + ...

RELATED SERIES.

A(x/2) appears to be an integer series:

A(x/2) = 1 + 2*x + 8*x^2 + 50*x^3 + 376*x^4 + 3124*x^5 + 27804*x^6 + 260496*x^7 + 2539616*x^8 + 25556330*x^9 + 263922884*x^10 + 2785341186*x^11 + ...

PROG

(PARI) {a(n) = my(A=[1]);

for(i=1, n, A=concat(A, 0); A[#A] = polcoeff(sum(m=0, #A, (-x)^m*(Ser(A)^m + (-1)^m)^m ), #A)); A[n+1]}

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

CROSSREFS

Sequence in context: A195193 A203435 A005263 * A113131 A195762 A127670

Adjacent sequences:  A325571 A325572 A325573 * A325575 A325576 A325577

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 16 2019

STATUS

approved

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Last modified December 7 00:16 EST 2019. Contains 329812 sequences. (Running on oeis4.)