A370440 -id:A370440 - cp's OEIS frontend

A264228 G.f. A(x) satisfies: A(x)^3 = A( x^3/(1-3*x) ), with A(0) = 0.

1, 1, 2, 5, 13, 35, 97, 274, 785, 2275, 6656, 19630, 58295, 174175, 523238, 1579584, 4789919, 14584723, 44577799, 136732988, 420784888, 1298937282, 4021383654, 12483820395, 38853994422, 121220646116, 379062880051, 1187912517953, 3730305167438, 11736596024002, 36994041916973, 116807229667919, 369415244627269, 1170113816365089

Offset: 1

Paul D. Hanna, Nov 08 2015

nonn

Radius of convergence is r = (sqrt(13) - 3)/2, where r = r^3/(1-3*r), with A(r) = 1.

Compare to a g.f. M(x) of Motzkin numbers: M(x)^2 = M(x^2/(1-2*x)) where M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x).

			G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 35*x^6 + 97*x^7 + 274*x^8 + 785*x^9 + 2275*x^10 + 6656*x^11 + 19630*x^12 + 58295*x^13 + 174175*x^14 + ...
where A(x)^3 = A( x^3/(1-3*x) ).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 28*x^6 + 87*x^7 + 270*x^8 + 839*x^9 + 2610*x^10 + 8127*x^11 + 25331*x^12 + 79035*x^13 + 246852*x^14 + 771808*x^15 + ...
A( x/(1+x+x^2) ) = x + x^4 + 2*x^7 + 6*x^10 + 22*x^13 + 88*x^16 + 367*x^19 + 1570*x^22 + 6843*x^25 + 30271*x^28 + 135530*x^31 + 612852*x^34 + 2794412*x^37 + 12832472*x^40 + ...
Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
B(x) = 1 + x + x^2 + x^3 - x^5 - x^6 + 2*x^8 + 3*x^9 - 6*x^11 - 9*x^12 + 20*x^14 + 30*x^15 - 71*x^17 - 110*x^18 + 267*x^20 + 419*x^21 - 1041*x^23 + ...
Let C0(x) and C2(x) be series trisections of B(x), B(x) = C0(x) + x + C2(x):
C0(x) = 1 + x^3 - x^6 + 3*x^9 - 9*x^12 + 30*x^15 - 110*x^18 + 419*x^21 - 1648*x^24 + 6652*x^27 - 27369*x^30 + 114384*x^33 - 484276*x^36 + ...
C2(x) = x^2 - x^5 + 2*x^8 - 6*x^11 + 20*x^14 - 71*x^17 + 267*x^20 - 1041*x^23 + 4168*x^26 - 17047*x^29 + 70902*x^32 + ... + (-1)^(n-1)*A370446(n)*x^(3*n-1) + ...
then C0(x) = x^2/C2(x).

Paul D. Hanna, Table of n, a(n) for n = 1..600

Cf. A264229, A264230, A361763, A370440, A370446.

{a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^3/(1-3*x +x*O(x^n))) )^(1/3) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.
(1) A(x)^3 = A( x^3/(1-3*x) ).
(2) A( x/(1+3*x) )^3 = A( x^3/(1+3*x)^2 ). - Paul D. Hanna, Mar 25 2023
(3) A( x/(1+x+x^2) )^3 = A( x^3/(1-x^3)^2 ). - Paul D. Hanna, Mar 11 2024

A370446 Expansion of g.f. A(x) satisfying A(x)^3 + x^4/A(x)^3 = A(x^3) + x^4/A(x^3) - 3*x^2.

Original entry on oeis.org

1, 1, 2, 6, 20, 71, 267, 1041, 4168, 17047, 70902, 298967, 1275141, 5491504, 23846271, 104295430, 459023543, 2031459236, 9034769573, 40358643042, 180998556943, 814645632727, 3678542796070, 16659932961647, 75657738747396, 344446195875766, 1571786529601990, 7187790264787872

Offset: 1

Paul D. Hanna, Mar 09 2024

nonn

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 71*x^6 + 267*x^7 + 1041*x^8 + 4168*x^9 + 17047*x^10 + 70902*x^11 + 298967*x^12 + 1275141*x^13 + 5491504*x^14 + 23846271*x^15 + ...
RELATED SERIES.
We can illustrate the formulas with the following related expansions.
(1) A(x)^3 + 2*x^2 + x^4/A(x)^3 = x - x^2 + x^3 - x^4 + x^6 - x^7 + 2*x^9 - 3*x^10 + 6*x^12 - 9*x^13 + 20*x^15 - 30*x^16 + 71*x^18 - 110*x^19 + 267*x^21 + ...
which equals A(x^3) - x^2 + x^4/A(x^3), as can be seen from
x^4/A(x^3) = x - x^4 - x^7 - 3*x^10 - 9*x^13 - 30*x^16 - 110*x^19 + ...
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 114*x^7 + 435*x^8 + 1715*x^9 + ...
x^4/A(x)^3 = x - 3*x^2 - 4*x^4 - 9*x^5 - 30*x^6 - 115*x^7 - 435*x^8 - 1713*x^9 + ...
(2) Let F(x) be the g.f. of A370440, which begins
F(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 15*x^7 + 30*x^8 + 55*x^9 + 113*x^10 + 274*x^11 + 683*x^12 + ...
where F(x)^3 = F( x^3 + 3*x^2*F(x)^2 ),
then the series reversion of F(x) begins
A(x^3) - x^2 + x^4/A(x^3) = x - x^2 + x^3 - x^4 + x^6 - x^7 + 2*x^9 - 3*x^10 + 6*x^12 - 9*x^13 + 20*x^15 - 30*x^16 + 71*x^18 - 110*x^19 + 267*x^21 - 419*x^22 + 1041*x^24 + ...
(3) Let G(x) be the g.f. of A264228, which begins
G(x) = x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 35*x^6 + 97*x^7 + 274*x^8 + 785*x^9 + 2275*x^10 + 6656*x^11 + 19630*x^12 +...
where G(x)^3 = G( x^3/(1 - 3*x) ),
then the series reversion of G(x) begins
-x^2/(A(-x^3) - x^2 + x^4/A(-x^3)) = x^2/(x + x^2 + x^3 + x^4 - x^6 - x^7 + 2*x^9 + 3*x^10 - 6*x^12 - 9*x^13 + 20*x^15 + 30*x^16 - 71*x^18 - 110*x^19 + 267*x^21 + 419*x^22 - 1041*x^24 +...).
SPECIFIC VALUES.
A(1/4.834464) = 0.349644497578571280258023712232522068793519739...
A(1/5) = 0.29940801195429552263938628184744484915469836164855...
A(1/6) = 0.21539123666426270273178791857213676628593723946879...
A(1/7) = 0.17414937372444126736977770687571455113383911571251...
A(1/8) = 0.14713126344900776621336355426627444003268957268553...
A(1/5^3) = 0.00806504925055020701973761348380106375185943151538...
A(1/6^3) = 0.00465126435780731657600811126033650347236250831668...
A(1/7^3) = 0.00292400175440295890949208907819991271975334925594...
which may be used to verify that the formula
A(x)^3 + x^4/A(x)^3 = A(x^3) + x^4/A(x^3) - 3*x^2
holds for these specific values.

Paul D. Hanna, Table of n, a(n) for n = 1..800

Cf. A370440, A264228.

{a(n) = my(A=x); for(m=1,n, A=truncate(A) +x^4*O(x^m); A = ( x^4/(x^4/subst(A,x,x^3) + subst(A,x,x^3) - A^3 - 3*x^2) +x^4*O(x^n))^(1/3) );polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 + 2*x^2 + x^4/A(x)^3 = A(x^3) - x^2 + x^4/A(x^3).
(2) F( A(x^3) - x^2 + x^4/A(x^3) ) = x, where F(x) = F( x^3 + 3*x^2*F(x)^2 )^(1/3) is the g.f. of A370440.
(3) G( -x^2/(A(-x^3) - x^2 + x^4/A(-x^3)) ) = x, where G(x) = G( x^3/(1 - 3*x) )^(1/3) is the g.f. of A264228.
a(n) ~ c * d^n / n^(3/2), where d = 4.8344630246454026903035642546835542141482126303313357979263... and c = 0.0713578385738499677445741870058758452888939567284935382... - Vaclav Kotesovec, Mar 13 2024
The radius of convergence r = 0.20684820525095397... = 1/d (where d is given above), and A(r) = 0.3497581458819115559285308998459940399916633464611700768... satisfy A(r) = r^(2/3) and A(r^3) = (5 - sqrt(21))/2 * r^2. - Paul D. Hanna, Mar 13 2024

A371709 Expansion of g.f. A(x) satisfying A( xA(x)^2 + xA(x)^3 ) = A(x)^3.

Original entry on oeis.org

1, 1, 1, 2, 6, 16, 39, 99, 271, 764, 2157, 6128, 17658, 51534, 151635, 448962, 1337493, 4008040, 12072594, 36524898, 110943633, 338218626, 1034509917, 3173811240, 9763898994, 30113782641, 93094164244, 288415278638, 895332445053, 2784580242557, 8675408291598, 27072326322939

Offset: 1

Paul D. Hanna, May 02 2024

nonn

Compare to the following identities of the Catalan function C(x) = x + C(x)^2 (A000108):
(1) C(x)^2 = C( x*C(x)*(1 + C(x)) ),
(2) C(x)^4 = C( x*C(x)^3*(1 + C(x))*(1 + C(x)^2) ),
(3) C(x)^8 = C( x*C(x)^7*(1 + C(x))*(1 + C(x)^2)*(1 + C(x)^4) ),
(4) C(x)^(2^n) = C( x*C(x)^(2^n-1)*Product_{k=0..n-1} (1 + C(x)^(2^k)) ) for n > 0.
a(3^n) == 1 (mod 3) for n >= 0.
a(2*3^n) == 1 (mod 3) for n >= 0.
a(n) == 2 (mod 3) iff n is the sum of 2 distinct powers of 3 (A038464).

			G.f. A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 16*x^6 + 39*x^7 + 99*x^8 + 271*x^9 + 764*x^10 + 2157*x^11 + 6128*x^12 + 17658*x^13 + 51534*x^14 + 151635*x^15 + 448962*x^16 + ...
where A( x*A(x)^2*(1 + A(x)) ) = A(x)^3.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 17*x^6 + 48*x^7 + 126*x^8 + 332*x^9 + 918*x^10 + 2616*x^11 + 7504*x^12 + ...
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 36*x^7 + 105*x^8 + 292*x^9 + 801*x^10 + 2256*x^11 + 6515*x^12 + 18981*x^13 + ...
A(x)^2 + A(x)^3 = x^2 + 3*x^3 + 6*x^4 + 12*x^5 + 30*x^6 + 84*x^7 + 231*x^8 + 624*x^9 + 1719*x^10 + 4872*x^11 + 14019*x^12 + 40599*x^13 + ...
Let B(x) be the series reversion of g.f. A(x), B(A(x)) = x, then
B(x) * (1+x)/(1+x^3) = x - 2*x^4 + 3*x^7 - 5*x^10 + 7*x^13 - 9*x^16 + 12*x^19 - 15*x^22 + 18*x^25 - 23*x^28 + ... + (-1)^n*A005704(n)*x^(3*n+1) + ...
where A005704 is the number of partitions of 3*n into powers of 3.
We can show that g.f. A(x) = A( x*A(x)^2*(1 + A(x)) )^(1/3) satisfies
(4) A(x) = x * Product_{n>=0} (1 + A(x)^(3^n))
by substituting x*A(x)^2*(1 + A(x)) for x in (4) to obtain
A(x)^3 = x * A(x)^2*(1 + A(x)) * Product_{n>=1} (1 + A(x)^(3^n))
which is equivalent to formula (4).
SPECIFIC VALUES.
A(3/10) = 0.526165645044542830201162330432965674027415264612114520...
A(1/4) = 0.353259384374080248921564026412797625837830114153200664...
A(1/5) = 0.255218141344695821239609680309162895225297482063273545...
A(t) = 1/2 and A(t*3/8) = 1/8 at t = (1/2)/Product_{n>=0} (1 + 1/2^(3^n)) = 0.295718718466711580562679377308518930409875701753934072...
A(t) = 1/3 and A(t*4/27) = 1/27 at t = (1/3)/Product_{n>=0} (1 + 1/3^(3^n)) = 0.241059181496179959557718992589733756750585121455883861...
A(t) = 1/4 and A(t*5/64) = 1/64 at t = (1/4)/Product_{n>=0} (1 + 1/4^(3^n)) = 0.196922325724019432212969925740117827612003158137366017...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A371716, A370440, A370441, A370446, A264228, A198951.

Cf. A005704, A038464.

/* Using series reversion of x/Product_{n>=0} (1 + x^(3^n)) */
{a(n) = my(A); A = serreverse( x/prod(k=0,ceil(log(n)/log(3)), (1 + x^(3^k) +x*O(x^n)) ) ); polcoeff(A,n)}
for(n=1,35, print1(a(n),", "))

/* Using A(x)^3 = A( x*A(x)^2 + x*A(x)^3 ) */
{a(n) = my(A=[1],F); for(i=1,n, A = concat(A,0); F = x*Ser(A);
A[#A] = polcoeff( subst(F,x, x*F^2 + x*F^3 ) - F^3, #A+2) ); A[n]}
for(n=1,35, print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = A( x*A(x)^2*(1 + A(x)) ).
(2) A(x)^9 = A( x*A(x)^8*(1 + A(x))*(1 + A(x)^3) ).
(3) A(x)^27 = A( x*A(x)^26*(1 + A(x))*(1 + A(x)^3)*(1 + A(x)^9) ).
(4) A(x) = x * Product_{n>=0} (1 + A(x)^(3^n)).
(5) A(x) = Series_Reversion( x / Product_{n>=0} (1 + x^(3^n)) ).
a(n) ~ c * d^n / n^(3/2), where d = 3.2753449994351908157330968510747739... and c = 0.1559869008021616116037651076359... - Vaclav Kotesovec, May 03 2024
The radius of convergence r of g.f. A(x) and A(r) satisfy 1 = Sum_{n>=0} 3^n * A(r)^(3^n) / (1 + A(r)^(3^n)) and r = A(r) / Product_{n>=0} (1 + A(r)^(3^n)), where r = 0.30531134893345362211... = 1/d (d is given above) and A(r) = 0.600582105427215700175254768411726892599... - Paul D. Hanna, May 03 2024

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.

cp's OEIS Frontend

A264228 G.f. A(x) satisfies: A(x)^3 = A( x^3/(1-3*x) ), with A(0) = 0.

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A370446 Expansion of g.f. A(x) satisfying A(x)^3 + x^4/A(x)^3 = A(x^3) + x^4/A(x^3) - 3*x^2.

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A371709 Expansion of g.f. A(x) satisfying A( x*A(x)^2 + x*A(x)^3 ) = A(x)^3.

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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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A371709 Expansion of g.f. A(x) satisfying A( xA(x)^2 + xA(x)^3 ) = A(x)^3.