A091190
G.f. A(x) satisfies x*A(x)^3 = B(x*A(x^3)) where B(x) = x/(1 - 3*x).
Original entry on oeis.org
1, 1, 2, 5, 13, 35, 97, 273, 778, 2240, 6499, 18976, 55703, 164243, 486130, 1443620, 4299365, 12836825, 38413933, 115184282, 346005073, 1041072108, 3137060983, 9465689545, 28596915843, 86492865522, 261876842801, 793661873276
Offset: 0
G.f.: A(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 + 55703*x^12 + ...
where A(x)^3 = A(x^3) / (1 - 3*x*A(x^3)).
RELATED SERIES.
A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 87*x^4 + 270*x^5 + 839*x^6 + 2607*x^7 + 8100*x^8 + 25169*x^9 + 78207*x^10 + 243009*x^11 + 755095*x^12 + ...
Also, D(x) = x*A(D(x)) is the g.f. of A370441, which begins
D(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 + ... + A370441(n)*x^n + ...
such that D(x)^3 = D( x^3 + 3*D(x)^4 ).
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m = 28; B[x_] = x/(1 - 3 x); A[_] = 1;
Do[A[x_] = (B[x A[x^3]]/x)^(1/3) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 29 2019 *)
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{a(n) = my(A,p=3,m=1); if(n<0,0, m=1; A=1+O(x); while(m<=n, m*=p; A = x*subst(A,x,x^p); A = (A/(1-p*A)/x)^(1/p)); polcoeff(A,n))}
for(n=0,30, print1(a(n),", "))
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.
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
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...
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{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),", "))
A371709
Expansion of g.f. A(x) satisfying A( x*A(x)^2 + x*A(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
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...
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/* 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),", "))
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/* 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),", "))
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