A361765 -id:A361765

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A361763

Expansion of g.f. A(x) satisfying A(x)^3 = A( x^3/(1 - 3*x)^3 ).

+10
5

1, 3, 9, 28, 93, 333, 1271, 5064, 20673, 85460, 355659, 1486719, 6238608, 26278281, 111114558, 471608944, 2008906581, 8586410085, 36816550550, 158332335279, 682843960665, 2952865525730, 12802463157570, 55646477022330, 242465061290160, 1059022767175173, 4636452916770489

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Related Catalan identity: F(x)^2 = F( x^2/(1 - 2*x)^2 ), where F(x) = x*C(x)^2 and C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Radius of convergence of g.f. A(x) is r where r is the real root of r = (1 - 3*r)^(3/2) with A(r) = 1 and r = (52 - (324*sqrt(717) + 8108)^(1/3) + (324*sqrt(717) - 8108)^(1/3))/162 = 0.214054846272632706742...

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..500

FORMULA

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:

(1) A(x)^3 = A( x^3/(1 - 3*x)^3 ).

(2) A(x^3) = A( x/(1 + 3*x) )^3.

(3) A(x) = x * Product_{n>=0} 1/(1 - 3/F(n,x))^(1/3^n), where F(0,x) = 1/x, F(m,x) = (F(m-1,x) - 3)^3 for m > 0.

(4) x/Series_Reversion(A(x)) = B(x) such that B(x)^3 = B(x^3) + 3*x (cf. A107092).

EXAMPLE

G.f.: A(x) = x + 3*x^2 + 9*x^3 + 28*x^4 + 93*x^5 + 333*x^6 + 1271*x^7 + 5064*x^8 + 20673*x^9 + 85460*x^10 + 355659*x^11 + 1486719*x^12 + ...

where

A( x^3/(1 - 3*x)^3 ) = x^3 + 9*x^4 + 54*x^5 + 273*x^6 + 1269*x^7 + 5670*x^8 + 24957*x^9 + 109593*x^10 + 482598*x^11 + 2133082*x^12 + ...

which equals A(x)^3.

RELATED SERIES.

Notice that the following cube root is an integer series

( A(x)/x )^(1/3) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 197*x^6 + 779*x^7 + 3135*x^8 + 12709*x^9 + 51757*x^10 + ... + A361762(n)*x^n + ...

Also, let B(x) satisfy A(x/B(x)) = x and B(A(x)) = A(x)/x,

then B(x) = x/Series_Reversion(A(x)) is the g.f. of A107092,

B(x) = 1 + 3*x + x^3 - x^6 + 2*x^9 - 4*x^12 + 9*x^15 - 22*x^18 + 55*x^21 - 142*x^24 + 376*x^27 - 1011*x^30 + ...

such that B(x)^3 = B(x^3) + 3*x,

as shown by the series

B(x)^(1/3) = 1 + x - x^2 + 2*x^3 - 4*x^4 + 9*x^5 - 22*x^6 + 55*x^7 - 142*x^8 + 376*x^9 - 1011*x^10 + ...

SPECIFIC VALUES.

A(1/5) = A(1/8)^(1/3) = 0.586384210523490911367880492498...

A(1/5) = (1/5) * (1 - 3/5)^(-1) * (1 - 3/8)^(-1/3) * (1 - 3/125)^(-1/9) * (1 - 3/1815848)^(-1/27) * ...

A(1/6) = A(1/27)^(1/3) = 0.346688997573685318336777346240...

A(1/6) = (1/6) * (1 - 3/6)^(-1) * (1 - 3/27)^(-1/3) * (1 - 3/13824)^(-1/9) * (1 - 3/2640087986661)^(-1/27) * ...

A(1/9) = A(1/216)^(1/3) = 0.16744549995321182031691216552466...

A(1/12) = A(1/729)^(1/3) = 0.11126394649161862248626102306202...

PROG

(PARI) {a(n) = my(A=x); for(i=1, #binary(n+1), A = ( subst(A, x, x^3/(1 - 3*x +x*O(x^n))^3 ) )^(1/3) ); polcoeff(A, n)}

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

CROSSREFS

Cf. A361762 ((A(x)/x)^(1/3)), A264230, A107092, A091190, A361765.

Cf. A264228, A264229, A264230.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 23 2023

STATUS

approved

A361764

Expansion of g.f. A(x) satisfying A(x)^5 = A( x^5/(1 - 5*x)^5 ) / (1 - 5*x).

+10
3

1, 1, 3, 11, 44, 185, 806, 3627, 16926, 82615, 425633, 2325804, 13438568, 81258283, 507109592, 3223435416, 20655599675, 132496854084, 847152571284, 5386490329194, 34026141582719, 213512516149309, 1331393810596499, 8255968489237781, 50955585198416275, 313329163267012645

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Related Catalan identity: C(x)^2 = C( x^2/(1 - 2*x)^2 ) / (1 - 2*x), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Radius of convergence of g.f. A(x) is r where r is the real root of r = (1 - 5*r)^(5/4) with A(r) = 1/r^(1/5) = 1.451902871451714... so that A(r)^5 = A(r)/(1 - 5*r) and r = 0.1549930338264677513709380922535...

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..500

FORMULA

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) A(x)^5 = A( x^5/(1 - 5*x)^5 ) / (1 - 5*x).

(2) A(x^5) = A( x/(1 + 5*x) )^5 / (1 + 5*x).

(3) A(x) = Product_{n>=1} 1/(1 - 5/F(n,x))^(1/5^n), where F(1,x) = 1/x, F(m,x) = (F(m-1,x) - 5)^5 for m > 1.

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 44*x^4 + 185*x^5 + 806*x^6 + 3627*x^7 + 16926*x^8 + 82615*x^9 + 425633*x^10 + ...

such that A(x)^5 = A( x^5/(1 - 5*x)^5 ) / (1 - 5*x).

RELATED SERIES.

A(x)^5 = 1 + 5*x + 25*x^2 + 125*x^3 + 625*x^4 + 3126*x^5 + 15655*x^6 + 78650*x^7 + 397625*x^8 + 2031875*x^9 + 10553128*x^10 + ...

A( x^5/(1 - 5*x)^5 ) = 1 + x^5 + 25*x^6 + 375*x^7 + 4375*x^8 + 43750*x^9 + 393753*x^10 + 3281400*x^11 + 25785375*x^12 + ...

SPECIFIC VALUES.

A(1/7) = ( 7/2 * A(1/32) )^(1/5) = 1.293495906485927953020670787280...

A(1/7) = (1 - 5/7)^(-1/5) * (1 - 5/32)^(-1/25) * (1 - 5/14348907)^(-1/125) * (1 - 5/14348902^5)^(-1/625) * ...

A(1/8) = ( 8/3 * A(1/243) )^(1/5) = 1.21774097368643014934892826038499995...

A(1/8) = (1 - 5/8)^(-1/5) * (1 - 5/243)^(-1/25) * (1 - 5/763633171168)^(-1/125) * (1 - 5/763633171163^5)^(-1/625) * ...

A(1/10) = ( 2 * A(1/3125) )^(1/5) = 1.14877193292427434012390599513357372...

A(1/10) = (1 - 5/10)^(-1/5) * (1 - 5/3125)^(-1/25) * (1 - 5/295646655283200000)^(-1/125) * (1 - 5/295646655283199995^5)^(-1/625) * ...

PROG

(PARI) {a(n) = my(A=1); for(i=1, #binary(n+1), A = ( subst(A, x, x^5/(1 - 5*x +x*O(x^n))^5 )/(1 - 5*x +x*O(x^n)) )^(1/5) ); polcoeff(A, n)}

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