A276358 -id:A276358 - cp's OEIS frontend

A291820 G.f. A(x,y) satisfies: A( x - xyA(x,y), y) = x + x(1-y)A(x,y), where the coefficients T(n,k) of x^n*y^k form a triangle read by rows n>=1, for k=0..n-1.

1, 1, 0, 1, 2, 0, 1, 7, 5, 0, 1, 16, 38, 14, 0, 1, 30, 157, 189, 42, 0, 1, 50, 477, 1245, 904, 132, 0, 1, 77, 1197, 5616, 8791, 4242, 429, 0, 1, 112, 2632, 19881, 55566, 57854, 19723, 1430, 0, 1, 156, 5250, 59327, 265204, 491947, 363880, 91366, 4862, 0, 1, 210, 9714, 155783, 1035442, 3062271, 4039551, 2220933, 423124, 16796, 0, 1, 275, 16929, 370205, 3472513, 15217674, 31979723, 31463341, 13285415, 1963169, 58786, 0, 1, 352, 28094, 811877, 10331673, 63678254, 197983540, 310618856, 235959185, 78419541, 9138416, 208012, 0

Offset: 1

Paul D. Hanna, Sep 01 2017

More generally, we have the following related identity.
Given functions F and G with F(0)=0, F'(0)=1, G(0)=0, G'(0)=0,
if F(x - y*G(x)) = x + (1-y)*G(x), then
(C1) F(x) = x + G( y*F(x) + (1-y)*x ),
(C2) y*F(x) + (1-y)*x = Series_Reversion(x - y*G(x)),
(C3) F(x) = x + G(x + y*G(x + y*G(x + y*G(x +...)))),
(C4) F(x) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) G(x)^n / n!.
The g.f. A(x,y) of this sequence equals F(x) in the above when G(x) = x*F(x).

			G.f.: A(x,y)  = x + x^2 + (2*y + 1)*x^3 + (5*y^2 + 7*y + 1)*x^4 +
(14*y^3 + 38*y^2 + 16*y + 1)*x^5 +
(42*y^4 + 189*y^3 + 157*y^2 + 30*y + 1)*x^6 +
(132*y^5 + 904*y^4 + 1245*y^3 + 477*y^2 + 50*y + 1)*x^7 +
(429*y^6 + 4242*y^5 + 8791*y^4 + 5616*y^3 + 1197*y^2 + 77*y + 1)*x^8 +
(1430*y^7 + 19723*y^6 + 57854*y^5 + 55566*y^4 + 19881*y^3 + 2632*y^2 + 112*y + 1)*x^9 +
(4862*y^8 + 91366*y^7 + 363880*y^6 + 491947*y^5 + 265204*y^4 + 59327*y^3 + 5250*y^2 + 156*y + 1)*x^10 +
(16796*y^9 + 423124*y^8 + 2220933*y^7 + 4039551*y^6 + 3062271*y^5 + 1035442*y^4 + 155783*y^3 + 9714*y^2 + 210*y + 1)*x^11 +
(58786*y^10 + 1963169*y^9 + 13285415*y^8 + 31463341*y^7 + 31979723*y^6 + 15217674*y^5 + 3472513*y^4 + 370205*y^3 + 16929*y^2 + 275*y + 1)*x^12 +...
such that
A( x - x*y*A(x,y), y)  =  x + x*(1-y)*A(x,y).
Also,
A(x,y) = x + Z*A(Z, y) where Z = y*A(x,y) + (1-y)*x.
...
This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) begins:
1;
1, 0;
1, 2, 0;
1, 7, 5, 0;
1, 16, 38, 14, 0;
1, 30, 157, 189, 42, 0;
1, 50, 477, 1245, 904, 132, 0;
1, 77, 1197, 5616, 8791, 4242, 429, 0;
1, 112, 2632, 19881, 55566, 57854, 19723, 1430, 0;
1, 156, 5250, 59327, 265204, 491947, 363880, 91366, 4862, 0;
1, 210, 9714, 155783, 1035442, 3062271, 4039551, 2220933, 423124, 16796, 0;
1, 275, 16929, 370205, 3472513, 15217674, 31979723, 31463341, 13285415, 1963169, 58786, 0;
1, 352, 28094, 811877, 10331673, 63678254, 197983540, 310618856, 235959185, 78419541, 9138416, 208012, 0;
1, 442, 44759, 1666522, 27896583, 232505790, 1014785477, 2355151627, 2859824058, 1721756609, 458956233, 42718416, 742900, 0; ...
RELATED SEQUENCES.
Given T(n,k) is the coefficient of x^n*y^k in g.f. A(x,y),
if b(n) = Sum_{k=0..n-1} T(n,k) * p^k * q^(n-k-1)
then B(x) = Sum_{n>=1} b(n)*x^n satisfies
(E1) B(x - p*x*B(x)) = x + (q-p)*x*B(x)
(E2) B(x)  =  x + Z*B(Z) where Z = p*B(x) + (q-p)*x.
...
G.f.s of columns of this triangle begin:
C.0: 1/(1-x)
C.1: (2 - x)/(1-x)^4
C.2: (5 + 3*x - 4*x^2 + x^3)/(1-x)^7
C.3: (14 + 49*x - 15*x^2 - 9*x^3 + 6*x^4 - x^5)/(1-x)^10
C.4: (42 + 358*x + 315*x^2 - 217*x^3 + 30*x^4 + 18*x^5 - 8*x^6 + x^7)/(1-x)^13
C.5: (132 + 2130*x + 5822*x^2 + 1403*x^3 - 1681*x^4 + 602*x^5 - 50*x^6 - 30*x^7 + 10*x^8 - x^9)/(1-x)^16
C.6: (429 + 11572*x + 62502*x^2 + 82763*x^3 + 2951*x^4 - 9760*x^5 + 5395*x^6 - 1329*x^7 + 75*x^8 + 45*x^9 - 12*x^10 + x^11)/(1-x)^19
C.7: (1430 + 59906*x + 541211*x^2 + 1506161*x^3 + 1217687*x^4 + 16416*x^5 - 35746*x^6 + 36682*x^7 - 13502*x^8 + 2550*x^9 - 105*x^10 - 63*x^11 + 14*x^12 - x^13)/(1-x)^22
C.8: (4862 + 301574*x + 4165915*x^2 + 19578410*x^3 + 34788033*x^4 + 20899306*x^5 + 1681742*x^6 + 174039*x^7 + 195964*x^8 - 103084*x^9 + 28953*x^10 - 4444*x^11 + 140*x^12 + 84*x^13 - 16*x^14 + x^15)/(1-x)^25
...
Thus A(x, y*(1-x)^3)*(1-x) = x + 2*y*x^3 + (5*y^2 - y)*x^4 + (14*y^3 + 3*y^2)*x^5 + (42*y^4 + 49*y^3 - 4*y^2)*x^6 + (132*y^5 + 358*y^4 - 15*y^3 + y^2)*x^7 +...

Paul D. Hanna, Table of n, a(n) for n = 1..1035, for rows 1..45 of the triangle in flattened form.

Cf. A088714 (row sums), A291821 (central terms), A291822 (diagonal).
Cf. A291813, A291814, A291815, A291816, A291817, A291818, A291819.
Cf. A276358, A291743, A291744.
Cf. A277295 (variant).

nmax = 13; A[x_] = x;
Do[A[x_] = x + (y A[x] + (1-y) x) A[y A[x] + (1-y) x] + x O[x]^n // Normal // Expand // Collect[#, x]&, {n, nmax}];
T[n_, k_] := SeriesCoefficient[A[x], {x, 0, n}, {y, 0, k}];
Table[T[n, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Oct 20 2019 *)

{T(n, k) = my(A=x); for(i=1, n, A = x + subst(x*A, x, y*A + (1-y)*x +x*O(x^n)) ); polcoeff(polcoeff(A, n, x), k, y)}
for(n=1, 12, for(k=0, n-1, print1(T(n, k), ", ")); print(""))

G.f. A(x,y) also satisfies:
(G1) A(x,y)  =  x + A( y*A(x,y) + x*(1-y), y).
(G2) y*A(x,y) + x*(1-y)  =  Series_Reversion( x - x*y*A(x,y) ).
(G3) x*y + (1-y)*B(x,y)  =  Series_Reversion( x + x*(1-y)*A(x,y) ), where B( A(x,y), y) = x.
(G4) A(x,y) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) A(x,y)^n * x^n / n!.
In formulas 2 and 3, the series reversion is taken with respect to variable x.
Formulas for terms:
(T1) T(n,0) = 1.
(T2) T(n,1) = (n-1)*n*(n+4)/6.  for n>=1.
(T3) T(n+1,n-1) =  binomial(2*n,n)/(n+1) = A000108(n) for n>=1.
Row sums:
(S1) Sum_{k=0..n-1} T(n,k) = A088714(n-1).
(S2) Sum_{k=0..n-1} T(n,k) * 2^(n-k-1) = A276358(n).
(S3) Sum_{k=0..n-1} T(n,k) * 3^(n-k-1) = A291744(n).
(S4) Sum_{k=0..n-1} T(n,k) * 2^k * 3^(n-k-1) = A291743(n).
(S5) Sum_{k=0..n-1} T(n,k) * 2^k = A291813(n).
(S6) Sum_{k=0..n-1} T(n,k) * 3^k = A291814(n).
(S7) Sum_{k=0..n-1} T(n,k) * 4^k = A291815(n).
(S8) Sum_{k=0..n-1} T(n,k) * 3^k * 2^(n-k-1) = A291816(n).
(S9) Sum_{k=0..n-1} T(n,k) * 3^k * 4^(n-k-1) = A291817(n).
(S10) Sum_{k=0..n-1} T(n,k) * 4^k * 3^(n-k-1) = A291818(n).
(S11) Sum_{k=0..n-1} T(n,k) * 4^(n-k-1) = A291819(n).

A291743 G.f. A(x) satisfies: A(x - 2xA(x)) = x + x*A(x).

Original entry on oeis.org

1, 3, 21, 213, 2649, 37683, 591345, 10018881, 180728985, 3438000843, 68495707461, 1421921135541, 30636666967857, 683015673332955, 15717420795468609, 372590181990474273, 9083851303291071633, 227458241320778287731, 5842792897560424283157, 153812115285865111498869, 4146016380695723021134761, 114341728157990237568733731, 3224107003868451067670682993

Offset: 1

Paul D. Hanna, Aug 30 2017

nonn

			G.f.: A(x) = x + 3*x^2 + 21*x^3 + 213*x^4 + 2649*x^5 + 37683*x^6 + 591345*x^7 + 10018881*x^8 + 180728985*x^9 + 3438000843*x^10 + 68495707461*x^11 + 1421921135541*x^12 +...
such that A(x - 2*x*A(x)) = x + x*A(x).
RELATED SERIES.
A(x - 2*x*A(x)) = x + x^2 + 3*x^3 + 21*x^4 + 213*x^5 + 2649*x^6 + 37683*x^7 + 591345*x^8 +...
which equals x + x*A(x).
Series_Reversion( x - x*A(x) ) = x + 2*x^2 + 14*x^3 + 142*x^4 + 1766*x^5 + 25122*x^6 + 394230*x^7 + 6679254*x^8 + 120485990*x^9 + 2292000562*x^10 + 45663804974*x^11 + 947947423694*x^12 +...
which equals (x + 2*A(x))/3.
A( (x + 2*A(x))/3 ) = x + 5*x^2 + 47*x^3 + 577*x^4 + 8273*x^5 + 131865*x^6 + 2275299*x^7 + 41821401*x^8 + 810261881*x^9 + 16426252597*x^10 + 346605967151*x^11 + 7582215510713*x^12 +...
which equals (A(x) - x) / (2*A(x) + x).

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A291744, A276358.

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

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

G.f. A(x) satisfies:
(1) A(x) = 3/2 * Series_Reversion( x - 2*x*A(x) ) - x/2.
(2) A(x) = x * (1 + A(B(x))) / (1 - 2*A(B(x))), where B(x) = (x + 2*A(x))/3.
(3) A( (x + 2*A(x))/3 )  =  (A(x) - x) / (2*A(x) + x).

A291744 G.f. A(x) satisfies: A(x - xA(x)) = x + 2x*A(x).

Original entry on oeis.org

1, 3, 15, 105, 897, 8739, 93663, 1080909, 13246017, 170728251, 2298619851, 32162768805, 465875706873, 6964550221215, 107193366978651, 1695277029466917, 27504875620268325, 457183442035485927, 7776605660061178251, 135234473290510961097, 2402252449086179775861, 43557766261735276367055, 805650777590230815177879, 15191845940176304945626737, 291896599103455803872483709, 5712079123789080942126760083

Offset: 1

Paul D. Hanna, Aug 30 2017

nonn

			G.f.: A(x) = x + 3*x^2 + 15*x^3 + 105*x^4 + 897*x^5 + 8739*x^6 + 93663*x^7 + 1080909*x^8 + 13246017*x^9 + 170728251*x^10 + 2298619851*x^11 + 32162768805*x^12 +...
such that A(x - x*A(x)) = x + 2*x*A(x).
RELATED SERIES.
A(x - x*A(x)) = x + 2*x^2 + 6*x^3 + 30*x^4 + 210*x^5 + 1794*x^6 + 17478*x^7 + 187326*x^8 + 2161818*x^9 + 26492034*x^10 + 341456502*x^11 + 4597239702*x^12 +...
which equals x + 2*x*A(x).
Series_Reversion( x - x*A(x) ) = x + x^2 + 5*x^3 + 35*x^4 + 299*x^5 + 2913*x^6 + 31221*x^7 + 360303*x^8 + 4415339*x^9 + 56909417*x^10 + 766206617*x^11 + 10720922935*x^12 +...
which equals (2*x + A(x))/3.
A( (2*x + A(x))/3 ) = x + 4*x^2 + 26*x^3 + 218*x^4 + 2126*x^5 + 22986*x^6 + 268410*x^7 + 3331482*x^8 + 43492370*x^9 + 592851806*x^10 + 8393229602*x^11 + 122922601030*x^12 +...
which equals (A(x) - x) / (A(x) + 2*x).

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A291743, A276358.

{a(n) = my(A=x); for(i=1, n, A = 3*serreverse( x - x*A +x*O(x^n) ) - 2*x ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

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

G.f. A(x) satisfies:
(1) A(x) = 3 * Series_Reversion( x - x*A(x) ) - 2*x.
(2) A(x) = x * (1 + 2*A(B(x))) / (1 - A(B(x))), where B(x) = (2*x + A(x))/3.
(3) A( (2*x + A(x))/3 )  =  (A(x) - x) / (A(x) + 2*x).

A291813 G.f. A(x) satisfies: A(x - 2xA(x)) = x - x*A(x).

Original entry on oeis.org

1, 1, 5, 35, 297, 2873, 30657, 353727, 4355497, 56709337, 775575269, 11085971235, 164979882033, 2548461481105, 40762085472929, 673751263927071, 11489101983573105, 201838769635965969, 3648620371959258149, 67795012307507004291, 1293607920940368319641, 25326486746707799668105, 508368313083167614599201, 10454499119633293760277151, 220120546753823908307191769, 4742197866143368618862457641

Offset: 1

Paul D. Hanna, Sep 01 2017

nonn

			G.f.: A(x) = x + x^2 + 5*x^3 + 35*x^4 + 297*x^5 + 2873*x^6 + 30657*x^7 + 353727*x^8 + 4355497*x^9 + 56709337*x^10 + 775575269*x^11 + 11085971235*x^12 +...
such that  A(x - 2*x*A(x)) = x - x*A(x).
RELATED SERIES.
A(x - 2*x*A(x)) = x - x^2 - x^3 - 5*x^4 - 35*x^5 - 297*x^6 - 2873*x^7 - 30657*x^8 +...
which equals x - x*A(x).
Series_Reversion( x - 2*x*A(x) ) = x + 2*x^2 + 10*x^3 + 70*x^4 + 594*x^5 + 5746*x^6 + 61314*x^7 + 707454*x^8 + 8710994*x^9 + 113418674*x^10 +...
which equals 2*A(x) - x.
A( 2*A(x) - x ) = x + 3*x^2 + 19*x^3 + 159*x^4 + 1561*x^5 + 17087*x^6 + 202975*x^7 + 2574391*x^8 + 34495545*x^9 + 484770627*x^10 + 7107406323*x^11 + 108289787415*x^12 + 1709478736593*x^13 + 27894511442079*x^14 +...
which equals (A(x) - x) / (2*A(x) - x).

Cf. A291820, A291814, A291815, A291816, A291817, A291818, A291819, A276358, A291743, A291744.

{a(n) = my(A=x); for(i=1, n, A = (1/2)*serreverse( x - 2*x*A +x*O(x^n) ) + x/2 ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

G.f. A(x) also satisfies:
(1) A(x) = (1/2)*Series_Reversion( x - 2*x*A(x) ) + x/2.
(2) A( 2*A(x) - x ) = (A(x) - x) / (2*A(x) - x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 2^k.

A291814 G.f. A(x) satisfies: A(x - 3xA(x)) = x - 2xA(x).

Original entry on oeis.org

1, 1, 7, 67, 769, 10009, 143359, 2218255, 36625657, 639659737, 11741022235, 225390779647, 4508109360985, 93665093491381, 2016669357747667, 44905700922069463, 1032419000661778213, 24472819932819733957, 597384952530618840715, 15000294032677574361955, 387082666821619977435277, 10256260095368150955828565, 278811213889895147327704519, 7770474960716476086765483619

Offset: 1

Paul D. Hanna, Sep 02 2017

nonn

			G.f.: A(x) = x + x^2 + 7*x^3 + 67*x^4 + 769*x^5 + 10009*x^6 + 143359*x^7 + 2218255*x^8 + 36625657*x^9 + 639659737*x^10 + 11741022235*x^11 + 225390779647*x^12 +...
such that  A(x - 3*x*A(x)) = x - 2*x*A(x).
RELATED SERIES.
A(x - 3*x*A(x)) = x - 2*x^2 - 2*x^3 - 14*x^4 - 134*x^5 - 1538*x^6 - 20018*x^7 +...
which equals x - 2*x*A(x).
Series_Reversion( x - 3*x*A(x) ) = x + 3*x^2 + 21*x^3 + 201*x^4 + 2307*x^5 + 30027*x^6 + 430077*x^7 + 6654765*x^8 +...
which equals 3*A(x) - 2*x.
A( 3*A(x) - 2*x ) = x + 4*x^2 + 34*x^3 + 382*x^4 + 5038*x^5 + 74134*x^6 + 1184650*x^7 + 20224990*x^8 + 364994554*x^9 + 6911857450*x^10 + 136622440786*x^11 + 2807805653098*x^12 +...
which equals (A(x) - x) / (3*A(x) - 2*x).

Cf. A291820, A291813, A291815, A291816, A291817, A291818, A291819, A276358, A291743, A291744.

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

G.f. A(x) also satisfies:
(1) A(x) = (1/3)*Series_Reversion( x - 3*x*A(x) ) + 2*x/3.
(2) A( 3*A(x) - 2*x ) = (A(x) - x) / (3*A(x) - 2*x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 3^k.

A291815 G.f. A(x) satisfies: A(x - 4xA(x)) = x - 3xA(x).

Original entry on oeis.org

1, 1, 9, 109, 1569, 25481, 454105, 8730373, 178996865, 3881556561, 88477557289, 2109927671453, 52443846331297, 1354646602217945, 36275862587452281, 1005099719255707829, 28765965099599741953, 849204340574458575777, 25827102287376124267593, 808349897942417046805197, 26011340193853765710238241, 859773626049480606121078057, 29168437337569276216572259097

Offset: 1

Paul D. Hanna, Sep 02 2017

nonn

			G.f.: A(x) = x + x^2 + 9*x^3 + 109*x^4 + 1569*x^5 + 25481*x^6 + 454105*x^7 + 8730373*x^8 + 178996865*x^9 + 3881556561*x^10 + 88477557289*x^11 + 2109927671453*x^12 +...
such that  A(x - 4*x*A(x)) = x - 3*x*A(x).
RELATED SERIES.
A(x - 4*x*A(x)) = x - 3*x^2 - 3*x^3 - 27*x^4 - 327*x^5 - 4707*x^6 - 76443*x^7 +...
which equals x - 3*x*A(x).
Series_Reversion( x - 4*x*A(x) ) = x + 4*x^2 + 36*x^3 + 436*x^4 + 6276*x^5 + 101924*x^6 + 1816420*x^7 + 34921492*x^8 +...
which equals 4*A(x) - 3*x.
A( 4*A(x) - 3*x )  = x + 5*x^2 + 53*x^3 + 741*x^4 + 12153*x^5 + 222405*x^6 + 4421501*x^7 + 93949493*x^8 + 2110952881*x^9 + 49786323589*x^10 + 1225967873349*x^11 + 31395927333829*x^12 +...
which equals (A(x) - x) / (4*A(x) - 3*x).

Cf. A291820, A291813, A291814, A291816, A291817, A291818, A291819, A276358, A291743, A291744.

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

G.f. A(x) also satisfies:
(1) A(x) = (1/4)*Series_Reversion( x - 4*x*A(x) ) + 3*x/4.
(2) A( 4*A(x) - 3*x) = (A(x) - x) / (4*A(x) - 3*x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 4^k.

A291816 G.f. A(x) satisfies: A(x - 3xA(x)) = x - x*A(x).

Original entry on oeis.org

1, 2, 16, 182, 2524, 39992, 699520, 13231034, 266985280, 5694001172, 127481465536, 2981125793144, 72532301230672, 1830526849868000, 47802726801684544, 1289123410465365782, 35841130838977837348, 1025903099063974343984, 30195807234087904770952, 912951678159786641659796, 28327442752528049524839856, 901289532361030971832330544, 29382621186595702051011638128

Offset: 1

Paul D. Hanna, Sep 02 2017

nonn

			G.f.: A(x) = x + 2*x^2 + 16*x^3 + 182*x^4 + 2524*x^5 + 39992*x^6 + 699520*x^7 + 13231034*x^8 + 266985280*x^9 + 5694001172*x^10 +...
such that  A(x - 3*x*A(x)) = x - x*A(x).
RELATED SERIES.
A(x - 3*x*A(x)) = x - x^2 - 2*x^3 - 16*x^4 - 182*x^5 - 2524*x^6 - 39992*x^7 - 699520*x^8 +...
which equals x - x*A(x).
Series_Reversion( x - 3*x*A(x) ) = x + 3*x^2 + 24*x^3 + 273*x^4 + 3786*x^5 + 59988*x^6 + 1049280*x^7 + 19846551*x^8 +...
which equals (3/2)*A(x) - x/2.
A( (3*A(x) - x)/2 )  = x + 5*x^2 + 52*x^3 + 713*x^4 + 11458*x^5 + 205160*x^6 + 3984304*x^7 + 82576109*x^8 + 1807215616*x^9 + 41461917398*x^10 +...
which equals (A(x) - x) / (3*A(x) - x).

Cf. A291820, A291813, A291814, A291815, A291817, A291818, A291819, A276358, A291743, A291744.

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

G.f. A(x) also satisfies:
(1) A(x) = (2/3)*Series_Reversion( x - 3*x*A(x) ) + x/3.
(2) A( (3*A(x) - x)/2 ) = (A(x) - x) / (3*A(x) - x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 3^k * 2^(n-k-1).

cp's OEIS Frontend

A291820 G.f. A(x,y) satisfies: A( x - x*y*A(x,y), y) = x + x*(1-y)*A(x,y), where the coefficients T(n,k) of x^n*y^k form a triangle read by rows n>=1, for k=0..n-1.

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A291743 G.f. A(x) satisfies: A(x - 2*x*A(x)) = x + x*A(x).

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A291744 G.f. A(x) satisfies: A(x - x*A(x)) = x + 2*x*A(x).

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A291813 G.f. A(x) satisfies: A(x - 2*x*A(x)) = x - x*A(x).

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A291814 G.f. A(x) satisfies: A(x - 3*x*A(x)) = x - 2*x*A(x).

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A291815 G.f. A(x) satisfies: A(x - 4*x*A(x)) = x - 3*x*A(x).

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A291816 G.f. A(x) satisfies: A(x - 3*x*A(x)) = x - x*A(x).

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A291820 G.f. A(x,y) satisfies: A( x - xyA(x,y), y) = x + x(1-y)A(x,y), where the coefficients T(n,k) of x^n*y^k form a triangle read by rows n>=1, for k=0..n-1.

A291743 G.f. A(x) satisfies: A(x - 2xA(x)) = x + x*A(x).

A291744 G.f. A(x) satisfies: A(x - xA(x)) = x + 2x*A(x).

A291813 G.f. A(x) satisfies: A(x - 2xA(x)) = x - x*A(x).

A291814 G.f. A(x) satisfies: A(x - 3xA(x)) = x - 2xA(x).

A291815 G.f. A(x) satisfies: A(x - 4xA(x)) = x - 3xA(x).

A291816 G.f. A(x) satisfies: A(x - 3xA(x)) = x - x*A(x).