A370446 -id:A370446 - 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

A370440 Expansion of g.f. A(x) satisfying A(x) = A( x^3 + 3x^2A(x)^2 )^(1/3), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 15, 30, 55, 113, 274, 683, 1596, 3547, 7990, 18968, 46530, 113663, 273392, 656421, 1598270, 3951520, 9827565, 24411649, 60599823, 150978177, 378293690, 951828992, 2398983638, 6051008950, 15284145261, 38690832455, 98154905623, 249390491237, 634296702273

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 + 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 + 1596*x^13 + 3547*x^14 + 7990*x^15 + ...
where A(x)^3 = A( x^3 + 3*x^2*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 18*x^7 + 47*x^8 + 106*x^9 + 216*x^10 + 450*x^11 + 1040*x^12 + ...
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 18*x^7 + 42*x^8 + 109*x^9 + 264*x^10 + 585*x^11 + 1270*x^12 + ...
Let B(x) denote the series reversion of A(x), A(B(x)) = x,
B(x) = 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 - 1648*x^25 + 4168*x^27 - 6652*x^28 + 17047*x^30 + ...
then B(x^3) = B(x)^3 + 3*x^2*B(x)^2, where
B(x)^2 = x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 3*x^6 - 3*x^8 + 4*x^9 - 8*x^11 + 11*x^12 - 23*x^14 + 34*x^15 + ...
B(x)^3 = x^3 - 3*x^4 + 6*x^5 - 10*x^6 + 12*x^7 - 9*x^8 + x^9 + 9*x^10 - 12*x^11 - x^12 + 24*x^13 - 33*x^14 + 69*x^16 - 102*x^17 + ...
Further, the trisections of B(x) = C1(x) + C2(x) + C3(x) are
C1(x) = x^4/C3(x) = x - x^4 - x^7 - 3*x^10 - 9*x^13 - 30*x^16 - 110*x^19 - ...
C2(x) = -x^2, and
C3(x) = x^3 + x^6 + 2*x^9 + 6*x^12 + 20*x^15 + 71*x^18 + 267*x^21 + 1041*x^24 + 4168*x^27 + 17047*x^30 + 70902*x^33 + ... + A370446(n)*x^(3*n) + ...
Compare these series to the series trisections involved in series reversion of A264228.
SPECIFIC VALUES.
A(1/3) = 0.5339969110985873619406256103732700685272...
A(1/4) = 0.3373018860609501862067597141160425025580...
A(1/5) = 0.2509433336474255853462277222741392614966...
A(1/6) = 0.2003115176013404351183299069966738623357...
A(1/8) = 0.1429156905534693639298206599148805278651...
A(1/3)^3 = A(1/27 + 3*A(1/3)^2/9) = A(0.132087937391...) = 0.152270661558...
A(1/4)^3 = A(1/64 + 3*A(1/4)^2/16) = A(0.036957355438...) = 0.038375699859...
A(1/5)^3 = A(1/125 + 3*A(1/5)^2/25) = A(0.015556706804...) = 0.250943333647...

Paul D. Hanna, Table of n, a(n) for n = 1..969

Cf. A370441, A370446, A264228, A356781.

{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 the following formulas.
(1) A(x) = A( x^3 + 3*x^2*A(x)^2 )^(1/3).
(2) B(x^3) = B(x)^3 + 3*x^2*B(x)^2, where A(B(x)) = x.
a(n) ~ c * d^n / n^(3/2), where d = 2.653503750287... and c = 0.193303... - Vaclav Kotesovec, Mar 14 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

A386659 G.f. A(x) satisfies A(x^3) = A(x)^3/(1 + 3*A(x)).

Original entry on oeis.org

1, 1, 0, 0, 1, 0, -1, 1, 0, -2, 3, 0, -6, 10, 0, -22, 33, 0, -79, 122, 0, -299, 472, 0, -1179, 1871, 0, -4754, 7601, 0, -19553, 31449, 0, -81720, 132020, 0, -345949, 561034, 0, -1480475, 2408712, 0, -6394189, 10431950, 0, -27835400, 45521500, 0, -122008360, 199948108, 0, -538016031, 883331845, 0

Offset: 1

Paul D. Hanna, Aug 28 2025

Compare to: F(x^2) = F(x)^2/(1 + 2*F(x)) holds when F(x) = x/(1-x).

			G.f.: A(x) = x + x^2 + x^5 - x^7 + x^8 - 2*x^10 + 3*x^11 - 6*x^13 + 10*x^14 - 22*x^16 + 33*x^17 - 79*x^19 + 122*x^20 - 299*x^22 + 472*x^23 - 1179*x^25 + 1871*x^26 - 4754*x^28 + ...
where A(x^3) = A(x)^3/(1 + 3*A(x)).
RELATED SERIES.
The series trisections are A(x) = T1(x) + T2(x) + T3(x), with T3(x) = 0 and
T1(x) = x - x^7 - 2*x^10 - 6*x^13 - 22*x^16 - 79*x^19 - 299*x^22 - 1179*x^25 - 4754*x^28 - 19553*x^31 - 81720*x^34 - 345949*x^37 - 1480475*x^40 + ...
T2(x) = x^2 + x^5 + x^8 + 3*x^11 + 10*x^14 + 33*x^17 + 122*x^20 + 472*x^23 + 1871*x^26 + 7601*x^29 + 31449*x^32 + 132020*x^35 + 561034*x^38 + 2408712*x^41 + ...
where T1(x)*T2(x) = A(x^3) and
T2(x)/T1(x) = x + x^4 + 2*x^7 + 6*x^10 + 20*x^13 + 71*x^16 + 267*x^19 + 1041*x^22 + 4168*x^25 + 17047*x^28 + ... + A370446(n)*x^(3*n-2) + ...
The cube of A(x) also has interesting series trisections.
A(x)^3 = x^3 + 3*x^4 + 3*x^5 + x^6 + 3*x^7 + 6*x^8 - 3*x^10 + 6*x^11 - 9*x^13 + 12*x^14 + x^15 - 21*x^16 + 42*x^17 - 84*x^19 + 132*x^20 - x^21 - 309*x^22 + 465*x^23 + x^24 + ...
where cubic trisections, defined by A(x)^3 = C1(x) + C2(x) + C3(x), obey
C3(x) = A(x^3),
C1(x)*C2(x) = 9*A(x^3)^3,
C2(x)/C1(x) = T2(x)/T1(x) = x + x^4 + 2*x^7 + 6*x^10 + 20*x^13 + 71*x^16 + 267*x^19 + 1041*x^22 + ... + A370446(n)*x^(3*n-2) + ...
The cubic trisections begin
C1(x) = 3*x^4 + 3*x^7 - 3*x^10 - 9*x^13 - 21*x^16 - 84*x^19 - 309*x^22 - 1137*x^25 - 4449*x^28 - 17868*x^31 - 73137*x^34 - 304662*x^37 - 1286388*x^40 - ...
C2(x) = 3*x^5 + 6*x^8 + 6*x^11 + 12*x^14 + 42*x^17 + 132*x^20 + 465*x^23 + 1791*x^26 + 7059*x^29 + 28503*x^32 + 117498*x^35 + 491757*x^38 + 2084481*x^41 + ...
C3(x) = x^3 + x^6 + x^15 - x^21 + x^24 - 2*x^30 + 3*x^33 - 6*x^39 + 10*x^42 - 22*x^48 + 33*x^51 + ... + a(n)*x^(3*n) + ...
SPECIFIC VALUES.
A(r) = 1 and A(r^3) = 1/4 at r = 0.591403538949431343296352603332310036448543950513103383318429...
A(t) = 4/5 and A(t^3) = 64/425 at t = 0.510303761967726164722767738473741580674762344121899...
A(t) = 3/4 and A(t^3) = 27/208 at t = 0.488075704869119285515484767956113771965332978558674...
A(t) = 2/3 and A(t^3) = 8/81 at t = 0.4490656139430636435247188510711544862057647445925319...
A(t) = 1/2 and A(t^3) = 1/20 at t = 0.3627219904933172573963798296372201737748692616169519...
A(t) = 1/3 and A(t^3) = 1/54 at t = 0.2629820536068200748031820994203659473004640287705972...
A(t) = 1/4 and A(t^3) = 1/112 at t = r^3 = 0.206848205250953970652722994332475597057157203674066...
A(t) = 1/5 and A(t^3) = 1/200 at t = 0.170714946526968286919515308872119424149511936479752...
A(1/2) = 0.7765855959847885627987696942587081429921785817514493... where A(1/8) = A(1/2)^3/(1 + 3*A(1/2)).
A(1/3) = 0.4482359377100401660271468423571796863698018480508060... where A(1/27) = A(1/3)^3/(1 + 3*A(1/3)).
A(1/4) = 0.3134295384970268001359461486249333443235800254018265... where A(1/64) = A(1/4)^3/(1 + 3*A(1/4)).
A(1/8) = 0.1406550988235082384593126468031209848166962450443705...
A(1/27) = 0.038408848749171730717291402355749106248762924579924...
A(1/64) = 0.015869141556098751959628853939856842544839850661716...

Paul D. Hanna, Table of n, a(n) for n = 1..2200

Cf. A370446, A264228.

{a(n) = my(V=[0,1]); for(i=1,n, V = concat(V,0); A = Ser(V);
V[#V] = polcoef( subst(A,x, x^3) - A^3/(1 + 3*A), #V+1)/3; ); V[n+1] }
for(n=1,54,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n along with series trisections T1(x) = Sum_{n>=0} a(3*n+1)*x^(3*n+1) and T2(x) = Sum_{n>=0} a(3*n+2)*x^(3*n+2) satisfy the following formulas.
(1) A(x^3) = A(x)^3/(1 + 3*A(x)).
(2) a(3*n) = 0 for n >= 1.
(3) T1(x)*T2(x) = A(x^3).
(4) T2(x)/T1(x) = G(x^3)/x^2 where g.f. G(x) of A370446 satisfies G(x)^3 + x^4/G(x)^3 = G(x^3) + x^4/G(x^3) - 3*x^2.
(5) A(-F(-x)) = x where g.f. F(x) of A264228 satisfies F(x)^3 = F( x^3/(1-3*x) ).

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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A370440 Expansion of g.f. A(x) satisfying A(x) = A( x^3 + 3*x^2*A(x)^2 )^(1/3), with A(0)=0, A'(0)=1.

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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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A386659 G.f. A(x) satisfies A(x^3) = A(x)^3/(1 + 3*A(x)).

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A370440 Expansion of g.f. A(x) satisfying A(x) = A( x^3 + 3x^2A(x)^2 )^(1/3), with A(0)=0, A'(0)=1.

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