A107092 -id:A107092 - cp's OEIS frontend

A361763 Expansion of g.f. A(x) satisfying A(x)^3 = A( x^3/(1 - 3*x)^3 ).

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

Offset: 1

Paul D. Hanna, Mar 23 2023

nonn

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...

			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...

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

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

Cf. A264228, A264229, A264230.

{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), ", "))

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).

A352702 G.f. A(x) satisfies: (1 - xA(x))^3 = 1 - 3x - x^3*A(x^3).

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 55, 142, 376, 1011, 2758, 7614, 21220, 59630, 168759, 480533, 1375676, 3957075, 11430582, 33144264, 96434321, 281447954, 823734157, 2417092933, 7109265120, 20955593252, 61893804180, 183148075432, 542885589115, 1611809502764, 4792612539375

Offset: 0

Paul D. Hanna, Mar 29 2022

nonn

Essentially an unsigned version of A107092 (after dropping the initial term).

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 22*x^5 + 55*x^6 + 142*x^7 + 376*x^8 + 1011*x^9 + 2758*x^10 + 7614*x^11 + ...
where
(1 - x*A(x))^3 = 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 + ...
also
(1 - 3*x - x^3*A(x^3))^(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 + ...
which equals 1 - x*A(x).

Paul D. Hanna, Table of n, a(n) for n = 0..1200
Ira M. Gessel, The Amdeberhan-Konvalinka Conjecture and Symmetric Functions, Séminaire Lotharingien Comb. (2024). See p. 83 of 109.

Cf. A107092, A352704, A352706.

{a(n) = my(A=1+x); for(i=1,n,
A = (1 - (1 - 3*x - x^3*subst(A,x,x^3) + x*O(x^(n+1)))^(1/3))/x);
polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) (1 + x*A(-x))^3 = 1 + 3*x + x^3*A(-x^3).
(2) A(x) = (1 - (1 - 3*x - x^3*A(x^3))^(1/3))/x.

A352703 G.f. A(x) satisfies: A(x)^5 = A(x^5) + 5*x.

Original entry on oeis.org

1, 1, -2, 6, -21, 80, -320, 1326, -5637, 24434, -107542, 479196, -2157045, 9792702, -44780606, 206055346, -953305632, 4431463863, -20686696920, 96931500840, -455722378776, 2149086843549, -10162544469252, 48176923330632, -228913129263389, 1089973058779915

Offset: 0

Paul D. Hanna, Mar 29 2022

sign

Not the same as A106223 or A196345.

			G.f.: A(x) = 1 + x - 2*x^2 + 6*x^3 - 21*x^4 + 80*x^5 - 320*x^6 + 1326*x^7 - 5637*x^8 + 24434*x^9 - 107542*x^10 + 479196*x^11 + ...
such that A(x)^5 = A(x^5) + 5*x, as illustrated by:
A(x)^5 = 1 + 5*x + x^5 - 2*x^10 + 6*x^15 - 21*x^20 + 80*x^25 - 320*x^30 + 1326*x^35 - 5637*x^40 + 24434*x^45 - 107542*x^50 + ...

Cf. A107092, A352704, A352705.

{a(n) = my(A=1+x); for(i=0,n\5,
A = (subst(A,x,x^5) + 5*x + x*O(x^(5*n)))^(1/5));
polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A352705 G.f. A(x) satisfies: A(x)^7 = A(x^7) + 7*x.

Original entry on oeis.org

1, 1, -3, 13, -65, 351, -1989, 11650, -69900, 427167, -2648438, 16612947, -105215448, 671760933, -4318468134, 27926126553, -181520036178, 1185220461867, -7769787812787, 51117085998498, -337373170647840, 2233091755252871, -14819626692452231, 98582852467595847

Offset: 0

Paul D. Hanna, Mar 29 2022

sign

Not the same as A106227.

			G.f.: A(x) = 1 + x - 3*x^2 + 13*x^3 - 65*x^4 + 351*x^5 - 1989*x^6 + 11650*x^7 - 69900*x^8 + 427167*x^9 - 2648438*x^10 + ...
such that A(x)^7 = A(x^7) + 7*x, as illustrated by:
A(x)^7 = 1 + 7*x + x^7 - 3*x^14 + 13*x^21 - 65*x^28 + 351*x^35 - 1989*x^42 + 11650*x^49 - 69900*x^56 + 427167*x^63 - 2648438*x^70 + ...

Cf. A107092, A352706, A352703.

{a(n) = my(A=1+x); for(i=1,n,
A = (subst(A,x,x^7) + 7*x + x*O(x^n))^(1/7));
polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A370441 Expansion of g.f. A(x) satisfying A(x) = A( x^3 + 3*A(x)^4 )^(1/3), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 1, 3, 12, 54, 261, 1324, 6952, 37461, 205977, 1151034, 6518085, 37321748, 215714904, 1256889150, 7374790400, 43537323406, 258417908640, 1541250594499, 9231988699115, 55514033703450, 334993491267955, 2027954403410504, 12312557796833622, 74955173794196890, 457431093085335708

Offset: 1

Paul D. Hanna, Mar 09 2024

nonn

Compare the g.f. to the following identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

			G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 54*x^5 + 261*x^6 + 1324*x^7 + 6952*x^8 + 37461*x^9 + 205977*x^10 + 1151034*x^11 + 6518085*x^12 + ...
where A(x)^3 = A( x^3 + 3*A(x)^4 ).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 55*x^6 + 270*x^7 + 1386*x^8 + 7347*x^9 + 39897*x^10 + 220779*x^11 + 1240392*x^12 + ...
A(x)^4 = x^4 + 4*x^5 + 18*x^6 + 88*x^7 + 451*x^8 + 2388*x^9 + 12958*x^10 + 71668*x^11 + 402489*x^12 + ...
Let B(x) denote the series reversion of A(x), A(B(x)) = x, where
B(x) = x - x^2 - x^3 - 2*x^4 - 4*x^5 - 9*x^6 - 22*x^7 - 55*x^8 - 142*x^9 - 376*x^10 - 1011*x^11 - 2758*x^12 + ... + (-1)^(n+1)*A107092(n)*x^n + ...
then B(x)^3 = B(x^3) - 3*x^4, where
B(x)^3 = x^3 - 3*x^4 - x^6 - x^9 - 2*x^12 - 4*x^15 - 9*x^18 - 22*x^21 - 55*x^24 - 142*x^27 - 376*x^30 - 1011*x^33 - 2758*x^36 + ...
Also, we have D(x) = x/B(x) is the g.f. of A091190, which begins
D(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + 97*x^6 + 273*x^7 + 778*x^8 + 2240*x^9 + 6499*x^10 + 18976*x^11 + ... + A091190(n)*x^n + ...
such that D(x)^3 = D(x^3)/(1 - 3*x*D(x^3)).

Cf. A107092, A091190, A370440.

{a(n) = my(A=[1],G); for(i=1,n, G = x*Ser(A); A = Vec((subst(G,x, x^3 + 3*x^2*G^2) + x^4*O(x^#A))^(1/3)); );A[n+1]}
for(n=0,40, print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n) * x^n satisfies:
(1) A(x) = A( x^3 + 3*A(x)^4 )^(1/3).
(2) B(x)^3 = B(x^3) - 3*x^4, where A(B(x)) = x.
(3) A(x) = x*D(A(x)) where D(x) = x/Series_Reversion(A(x)) is the g.f. of A091190.

A107093 G.f. A(x) satisfies: A(x) = A(x^3)^(1/3) + 3*x.

Original entry on oeis.org

1, 3, 0, 1, 0, 0, -1, 0, 0, 2, 0, 0, -4, 0, 0, 9, 0, 0, -22, 0, 0, 55, 0, 0, -142, 0, 0, 376, 0, 0, -1011, 0, 0, 2758, 0, 0, -7614, 0, 0, 21220, 0, 0, -59630, 0, 0, 168759, 0, 0, -480533, 0, 0, 1375676, 0, 0, -3957075, 0, 0, 11430582, 0, 0, -33144264, 0, 0, 96434321, 0, 0, -281447954, 0, 0, 823734157, 0, 0, -2417092933, 0, 0

Offset: 0

Paul D. Hanna, May 11 2005

sign

Self-convolution cube of A107092.

			A(x) = 1 + 3*x + x^3 - x^6 + 2*x^9 - 4*x^12 + 9*x^15 - 22*x^18 +...
A(x^3)^(1/3) = 1 + x^3 - x^6 + 2*x^9 - 4*x^12 + 9*x^15 - 22*x^18 +...

Cf. A107092.

{a(n)=local(A=1+x);for(i=1,n,A=(subst(A,x,x^3)+3*x+x*O(x^n))^(1/3)); polcoeff(A^3,n,x)}

cp's OEIS Frontend

A361763 Expansion of g.f. A(x) satisfying A(x)^3 = A( x^3/(1 - 3*x)^3 ).

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A352702 G.f. A(x) satisfies: (1 - x*A(x))^3 = 1 - 3*x - x^3*A(x^3).

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A352703 G.f. A(x) satisfies: A(x)^5 = A(x^5) + 5*x.

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A352705 G.f. A(x) satisfies: A(x)^7 = A(x^7) + 7*x.

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A370441 Expansion of g.f. A(x) satisfying A(x) = A( x^3 + 3*A(x)^4 )^(1/3), with A(0)=0, A'(0)=1.

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A107093 G.f. A(x) satisfies: A(x) = A(x^3)^(1/3) + 3*x.

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A352702 G.f. A(x) satisfies: (1 - xA(x))^3 = 1 - 3x - x^3*A(x^3).