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A276365 G.f. A(x) satisfies: A(x - 2*A(x)^2) = x - A(x)^2. 2
1, 1, 6, 53, 578, 7234, 100044, 1495125, 23802346, 399740086, 7032766196, 128952474242, 2454645604820, 48359400068836, 983683769369624, 20618782389897333, 444636132851851386, 9851377271964349038, 223998085060636514020, 5221799494107885481430, 124695762315403816775932, 3047952254964607540099676, 76206565881709345978097960, 1947752912315470845518308642, 50860833685759573411702643972 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..300

FORMULA

G.f. A(x) also satisfies:

(1) A(x) = x + A( 2*A(x) - x )^2.

(2) 2*A(x) = x + Series_Reversion(x - 2*A(x)^2).

(3) R(x) = 2*x - Series_Reversion(x - A(x)^2), where R(A(x)) = x.

(4) R( ( x - R(x) )^(1/2) ) = 2*x - R(x), where R(A(x)) = x.

(5) A(x) = x + Sum_{n>=1} 2^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.

a(n) = Sum_{k=0..n-1} A277295(n,k)*2^k.

EXAMPLE

G.f.: A(x) = x + x^2 + 6*x^3 + 53*x^4 + 578*x^5 + 7234*x^6 + 100044*x^7 + 1495125*x^8 + 23802346*x^9 + 399740086*x^10 + 7032766196*x^11 +...

such that A(x - 2*A(x)^2) = x - A(x)^2.

RELATED SERIES.

Note that Series_Reversion(x - 2*A(x)^2) = 2*A(x) - x, which begins:

Series_Reversion(x - 2*A(x)^2) = x + 2*x^2 + 12*x^3 + 106*x^4 + 1156*x^5 + 14468*x^6 + 200088*x^7 + 2990250*x^8 + 47604692*x^9 + 799480172*x^10 +...

Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,

R(x) = x - x^2 - 4*x^3 - 28*x^4 - 264*x^5 - 2992*x^6 - 38496*x^7 - 544464*x^8 - 8298080*x^9 - 134500672*x^10 - 2297361024*x^11 +...

then Series_Reversion(x - A(x)^2) = 2*x - R(x).

PROG

(PARI) {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - 2*F^2) + F^2, #A) ); A[n]}

for(n=1, 30, print1(a(n), ", "))

CROSSREFS

Cf. A277295, A213591, A275765.

Sequence in context: A055973 A223345 A066357 * A185148 A243921 A109092

Adjacent sequences:  A276362 A276363 A276364 * A276366 A276367 A276368

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 01 2016

STATUS

approved

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Last modified August 24 03:51 EDT 2019. Contains 326260 sequences. (Running on oeis4.)