A030266 -id:A030266 - cp's OEIS frontend

A128325 Rectangular table, read by antidiagonals, where the g.f.s of row n, R(x,n), satisfy: R(x,n+1) = R(G(x),n) for n>=0 and xR(x,0) = G(x) = x + xG(G(x)) is the g.f. of A030266.

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 12, 23, 1, 1, 5, 20, 57, 104, 1, 1, 6, 30, 114, 305, 531, 1, 1, 7, 42, 200, 712, 1787, 2982, 1, 1, 8, 56, 321, 1435, 4772, 11269, 18109, 1, 1, 9, 72, 483, 2608, 10900, 33896, 75629, 117545, 1, 1, 10, 90, 692, 4389, 22219, 86799

Offset: 0

Paul D. Hanna, Mar 11 2007

Row n equals 1 + (n+2)-th self-composition of the g.f. G(x) of A030266: R(x,0) = 1 + G(G(x)); R(x,1) = 1 + G(G(G(x))); R(x,2) = 1 + G(G(G(G(x)))); etc.

			Consider the infinite system of simultaneous equations:
  A = 1 + x*A*B;
  B = 1 + x*A*B*C;
  C = 1 + x*A*B*C*D;
  D = 1 + x*A*B*C*D*E;
  E = 1 + x*A*B*C*D*E*F; ...
The unique solution to the variables are:
  A = R(x,0), B = R(x,1), C = R(x,2), D = R(x,3), E = R(x,4), etc.,
where R(x,n) denotes the g.f. of row n of this table and satisfies:
  R(x,1) = R(x*A,0); R(x,2) = R(x*A,1); R(x,3) = R(x*A,2); etc.
The row g.f.s are also related by:
  R(x,0) = 1 + x/(1 - x*R(x,1) - x*R(x,2));
  R(x,1) = 1 + x/(1 - x*R(x,1) - x*R(x,2) - x*R(x,3));
  R(x,2) = 1 + x/(1 - x*R(x,1) - x*R(x,2) - x*R(x,3) - x*R(x,4)); etc.
The initial rows of this table begin:
  R(x,0): [1, 1,  2,   6,   23,   104,    531,    2982,    18109, ...];
  R(x,1): [1, 1,  3,  12,   57,   305,   1787,   11269,    75629, ...];
  R(x,2): [1, 1,  4,  20,  114,   712,   4772,   33896,   253102, ...];
  R(x,3): [1, 1,  5,  30,  200,  1435,  10900,   86799,   720074, ...];
  R(x,4): [1, 1,  6,  42,  321,  2608,  22219,  196910,  1805899, ...];
  R(x,5): [1, 1,  7,  56,  483,  4389,  41531,  406441,  4095749, ...];
  R(x,6): [1, 1,  8,  72,  692,  6960,  72512,  777888,  8559852, ...];
  R(x,7): [1, 1,  9,  90,  954, 10527, 119832, 1399755, 16720998, ...];
  R(x,8): [1, 1, 10, 110, 1275, 15320, 189275, 2392998, 30865353, ...];
  R(x,9): [1, 1, 11, 132, 1661, 21593, 287859, 3918189, 54301621, ...];
  R(x,10):[1, 1, 12, 156, 2118, 29624, 423956, 6183400, 91673594, ...]; ...

Cf. A030266 (row 0), A128326 (row 1), A128327 (row 2), A128328 (row 3), A128329 (main diagonal); A128330 (variant).

{T(n,k)=local(A=vector(n+k+3,m,1+x+x*O(x^(n+k)))); for(i=1,n+k+3,for(j=1,n+k+1,N=n+k+2-j; A[N]=1+x/(1-x*sum(m=2,N+2,A[m]+x*O(x^(n+k))))));Vec(A[n+1])[k+1]}

Let R(x,n) denote the g.f. of row n of this table, then
R(x,n) = 1 + x*Product_{k=0..n+1} R(x,k),
R(x,n) = 1 + x/[1 - x*Sum_{k=1..n+2} R(x,k) ].

A128326 G.f.: A(x) = 1 + G(G(G(x))), where G(x) = x + x*G(G(x)) is the g.f. of A030266.

Original entry on oeis.org

1, 1, 3, 12, 57, 305, 1787, 11269, 75629, 535960, 3987913, 31021693, 251445581, 2117993712, 18499513147, 167246537937, 1562556275281, 15066167302802, 149737897716757, 1532313152898208, 16129331500727047

Offset: 0

Paul D. Hanna, Mar 11 2007

nonn

Equals row 1 of table A128325.

Cf. A030266; A128325 (table), A128327 (row 2), A128328 (row 3), A128329 (main diagonal).

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

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

G.f. satisfies: A(x) = x/(1 - A( x/(1 - A(x)) )) when offset is taken to be 1. - Paul D. Hanna, Dec 20 2014

A125280 Row sums of triangle A125280, which is the convolution triangle of A030266.

Original entry on oeis.org

1, 2, 5, 15, 53, 217, 1011, 5260, 30041, 185677, 1228209, 8620874, 63792445, 495163451, 4015888557, 33923543492, 297706713081, 2708377382444, 25495655264883, 247952347547483, 2487743315817023, 25717746952124842

Offset: 0

Paul D. Hanna, Nov 26 2006

nonn

			A(x) = 1 + 2*x + 5*x^2 + 15*x^3 + 53*x^4 + 217*x^5 + 1011*x^6 +...
where 1 - 1/(1 + x*A(x)) = G(x) is the g.f. of A030266:
G(x) = x + x^2 + 2*x^3 + 6*x^4 + 23*x^5 + 104*x^6 + 531*x^7 + 2982*x^8+..

Cf. A125280, A030266.

{a(n)=local(G=x+x^2);for(i=0,n,G=x+x*subst(G,x,G+x^2*O(x^n))); polcoeff((-1+1/(1-G))/x,n,x)}

G.f.: A(x) = (1/x)*G(x)/(1 - G(x)) where G(x) = x + x*G(G(x)) is g.f. of A030266.

A125278 Convolution triangle of A030266, which shifts left under self-COMPOSE.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 5, 3, 1, 23, 16, 9, 4, 1, 104, 62, 31, 14, 5, 1, 531, 278, 123, 52, 20, 6, 1, 2982, 1398, 552, 213, 80, 27, 7, 1, 18109, 7718, 2750, 964, 340, 116, 35, 8, 1, 117545, 46083, 14976, 4784, 1561, 513, 161, 44, 9, 1, 808764, 294392, 88083, 25792, 7755

Offset: 0

Paul D. Hanna, Nov 26 2006

Column 0 of matrix square T^2 equals column 0 of T shift left. Central terms are T(2*n,n) = (n+1)*A125279(n).

			Triangle begins:
1;
1, 1;
2, 2, 1;
6, 5, 3, 1;
23, 16, 9, 4, 1;
104, 62, 31, 14, 5, 1;
531, 278, 123, 52, 20, 6, 1;
2982, 1398, 552, 213, 80, 27, 7, 1;
18109, 7718, 2750, 964, 340, 116, 35, 8, 1;
117545, 46083, 14976, 4784, 1561, 513, 161, 44, 9, 1;
808764, 294392, 88083, 25792, 7755, 2400, 742, 216, 54, 10, 1; ...
Matrix square T^2 begins:
1;
2, 1;
6, 4, 1;
23, 16, 6, 1;
104, 70, 30, 8, 1;
531, 336, 149, 48, 10, 1; ...
which is also a convolution triangle.

Cf. A030266, A125279, A125280 (row sums).

T(n,k)=if(n0,sum(j=0,n-k,T(j,0)*T(n-1-j,k-1)), sum(j=0,n-1,T(j,0)*T(n-1,j)))))

T(0,0) = 1 ; for n>0: T(n,0) = Sum_{j=0..n-1} T(j,0)*T(n-1,j) = A030266(n) (self-COMPOSE); for k>0: T(n,k) = Sum_{j=0..n-k} T(j,0)*T(n-1-j,k-1) (self-convolutions of A030266).

A125279 G.f.: A(x) = (1/x)*series_reversion(x^2/G(x)) where G(x) is the g.f. of A030266, which shifts left under self-COMPOSE.

Original entry on oeis.org

1, 1, 3, 13, 68, 400, 2555, 17375, 124280, 927711, 7189102, 57627044, 476645965, 4061184195, 35604795538, 320957712849, 2973550524004, 28305757130713, 276806230525768, 2780528226936569, 28686373905833717, 303913110837114965

Offset: 0

Paul D. Hanna, Nov 26 2006

nonn

Derived from central terms of triangle: a(n) = A125278(2*n,n)/(n+1).

			A(x) = 1 + x + 3*x^2 + 13*x^3 + 68*x^4 + 400*x^5 + 2555*x^6 +...
The g.f. of A030266 is G(x) = x + x*G(G(x)) where
G(x) = x + x^2 + 2*x^3 + 6*x^4 + 23*x^5 + 104*x^6 + 531*x^7 + 2982*x^8+..

Cf. A030266, A125278.

{a(n)=local(A=1+x);for(i=0,n,A=1+x*A^2*subst(A,x,x^2*A^4/(A-1+x^2*O(x^n)))); polcoeff(A,n,x)}

G.f. satisfies: A(x) = 1 + x*A(x)^2 * A( x^2*A(x)^4/(A(x) - 1) ). By definition, G.f. satisfies: A(x) = 1 + G(x*A(x)^2); G(x*A(x)) = x*A(x)^2; x*A(x^2/G(x)) = G(x); where G(x) = x + x*G(G(x)) is g.f. of A030266.

A141141 The main diagonal in the table of coefficients of iterations of G(x), where G(x) = x + x*G(G(x)) = g.f. of A030266.

Original entry on oeis.org

1, 2, 12, 114, 1435, 22219, 406441, 8559852, 203792337, 5409449156, 158350300141, 5066765087000, 175908765569628, 6585443884172129, 264428161094825151, 11335716352419699208, 516717363793695685925, 24955728581736822645816

Offset: 1

Paul D. Hanna, Jun 10 2008

nonn

			a(n) = the n-th coefficient of the n-th iteration of G(x):
[x] G(x) = 1, [x^2] G(G(x)) = 2, [x^3] G(G(G(x))) = 12, etc.
The initial iterations (n=1..7) of G(x) are:
n=1: x + x^2 + 2*x^3 + 6*x^4 + 23*x^5 + 104*x^6 + 531*x^7 +...
n=2: x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 531*x^6 + 2982*x^7 +...
n=3: x + 3*x^2 + 12*x^3 + 57*x^4 + 305*x^5 + 1787*x^6 + 11269*x^7 +...
n=4: x + 4*x^2 + 20*x^3 + 114*x^4 + 712*x^5 + 4772*x^6 + 33896*x^7 +...
n=5: x + 5*x^2 + 30*x^3 + 200*x^4 + 1435*x^5 + 10900*x^6 + 86799*x^7+...
n=6: x + 6*x^2 + 42*x^3 + 321*x^4 + 2608*x^5 + 22219*x^6 + 196910*x^7+...
n=7: x + 7*x^2 + 56*x^3 + 483*x^4 + 4389*x^5 + 41531*x^6 + 406441*x^7+...
Notice the main diagonal of the table formed from these coefficients.

Cf. A030266, A128325, A128329.

{a(n)=local(A=x,B); if(n<1, 0, for(i=1, n, A=serreverse(x/(1+A +x*O(x^n)))); B=x;for(i=1,n,B=subst(A,x,B+x*O(x^n)));polcoeff(B,n))}

A088714 G.f. satisfies A(x) = 1 + xA(x)^2A(x*A(x)).

Original entry on oeis.org

1, 1, 3, 13, 69, 419, 2809, 20353, 157199, 1281993, 10963825, 97828031, 907177801, 8716049417, 86553001779, 886573220093, 9351927111901, 101447092428243, 1130357986741545, 12923637003161409, 151479552582252239

Offset: 0

Paul D. Hanna, Oct 12 2003, May 22 2008

nonn

Equals row sums of triangle A291820.

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 419*x^5 + 2809*x^6 +...
The g.f. A(x) satisfies:
x*A(x) = x + x^2*A(x) + d/dx x^4*A(x)^2/2! + d^2/dx^2 x^6*A(x)^3/3! + d^3/dx^3 x^8*A(x)^4/4! +...
The logarithm of the g.f. is given by:
log(A(x)) = x*A(x) + d/dx x^3*A(x)^2/2! + d^2/dx^2 x^5*A(x)^3/3! + d^3/dx^3 x^7*A(x)^4/4! + d^4/dx^4 x^9*A(x)^5/5! +...
From _Paul D. Hanna_, Apr 16 2007: (Start)
G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x*A*B;
B = A + x*B*C;
C = B + x*C*D;
D = C + x*D*E;
E = D + x*E*F ; ...
where variables B,C,D,E,..., are formed from successive iterations of x*A(x):
B = A(x)*A(x*A(x)), C = B*A(x*B), D = C*A(x*C), E = D*A(x*D), ...;
more explicilty,
B = 1 + 2*x + 8*x^2 + 42*x^3 + 258*x^4 + 1764*x^5 + 13070*x^6 +...,
C = 1 + 3*x + 15*x^2 + 93*x^3 + 655*x^4 + 5039*x^5 + 41453*x^6 +...,
D = 1 + 4*x + 24*x^2 + 172*x^3 + 1372*x^4 + 11796*x^5 +...,
E = 1 + 5*x + 35*x^2 + 285*x^3 + 2545*x^4 + 24255*x^5 +...,
... (End)
Related expansions:
A(x*A(x)) = 1 + x + 4*x^2 + 22*x^3 + 142*x^4 + 1016*x^5 + 7838*x^6 + 64174*x^7 + 552112*x^8 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 173*x^4 + 1054*x^5 + 7039*x^6 + 50632*x^7 + 387613*x^8 +...
d/dx x^4*A(x)^2/2! = 2*x^3 + 5*x^4 + 21*x^5 + 112*x^6 + 692*x^7 + 4743*x^8 +...
d^2/dx^2 x^6*A(x)^3/3! = 5*x^4 + 21*x^5 + 112*x^6 + 696*x^7 + 4815*x^8 +...
d^3/dx^3 x^8*A(x)^4/4! = 14*x^5 + 84*x^6 + 540*x^7 + 3795*x^8 +...
d^4/dx^4 x^10*A(x)^5/5! = 42*x^6 + 330*x^7 + 2475*x^8 + 19305*x^9 +...
...
d^(n-1)/dx^(n-1) x^(2*n)*A(x)^n/n! = A000108(n)*x^(n+1) +...

Cf. A291820, A088713, A212910, A212919.
Apart from signs, same as A067145. - Philippe Deléham, Jun 18 2006
Cf. A002449, A030266, A087949, A088717, A091713, A120971.

m = 21; A[] = 1; Do[A[x] = 1 + x A[x]^2 A[x A[x]] + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 06 2019 *)

{a(n) = my(A); if( n<0, 0, n++; A = x + O(x^2); for(i=2, n, A = x / (1 - subst(A, x, A))); polcoeff(A, n))}; /* Michael Somos, May 21 2005 */

{a(n)=local(A); if(n<0, 0, A=1+x+O(x^2); for(i=1,n, A=1/(1-x*A*subst(A,x,x*A)));polcoeff(A,n))}

{a(n)=local(A); if(n<0, 0, A=1+x+O(x^2);for(i=0,n, A=(1/x)*serreverse(x-x^2*A));polcoeff(A,n))}

{a(n,m=1)=if(n==0,1,if(m==0,0^n,sum(k=0,n,m*binomial(n+k+m,k)/(n+k+m)*a(n-k,k))))} \\ Paul D. Hanna, Jul 09 2009

/* n-th Derivative: */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
/* G.f.: [Paul D. Hanna, Dec 18 2010] */
{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n,A=exp(sum(m=1, n, Dx(m-1, x^(2*m-1)*A^m/m!))+x*O(x^n))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* n-th Derivative: */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
/* G.f.: [Paul D. Hanna, May 31 2012] */
{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n,A=1+(1/x)*sum(m=1, n+1, Dx(m-1, x^(2*m)*A^m/m!))+x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

G.f. satisfies:
(1) A(x) = (1/x)*Series_Reversion(x - x^2*A(x)).
(2) A(x) = 1 + (1/x)*Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n)*A(x)^n/n!.
(3) A(x) = exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^n/n! ).
(4) A(x) = 1/(1 - x*A(x)*A(x*A(x))).
(5) A(x) = f(x*A(x)) = (1-1/f(x))/x where f(x) is the g.f. of A088713.
Given g.f. A(x), then B(x) = x*A(x) satisfies 0 = f(x, B(x), B(B(x))) where f(a0, a1, a2) = a0 - a1 + a1*a2. - Michael Somos, May 21 2005
From Paul D. Hanna, Jul 09 2009: (Start)
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then
a(n,m) = Sum_{k=0..n} m*C(n+k+m,k)/(n+k+m) * a(n-k,k).
(End)
a(n) = Sum_{k=0..n} A291820(n+1,k). - Paul D. Hanna, Sep 01 2017

A087949 G.f. satisfies A(x) = 1 + xA(xA(x)).

Original entry on oeis.org

1, 1, 1, 2, 5, 16, 59, 246, 1131, 5655, 30428, 174835, 1066334, 6870542, 46581883, 331237074, 2463361903, 19112314727, 154364077009, 1295325828045, 11273167827343, 101589943242179, 946577526626181, 9107029927925714, 90359115887726302, 923509462029444933

Offset: 0

Paul D. Hanna, Sep 16 2003

nonn

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 16*x^5 +...
A(xA(x)) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 59*x^5 +...
Logarithmic series:
log(A(x)) = x/A(x) + [d/dx x^3*A(x)^2]*A(x)^(-4)/2! + [d^2/dx^2 x^5*A(x)^3]*A(x)^(-6)/3! + [d^3/dx^3 x^7*A(x)^4]*A(x)^(-8)/4! +...
Let G(x) = x*A(x) then
x = G(x*[1 - G(x) + 2*G(x)^2 - 5*G(x)^3 + 14*G(x)^4 - 42*G(x)^5 +-...])
where the unsigned coefficients are the Catalan numbers (A000108).

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A002449, A030266, A088714, A088717, A091713, A120971, A140092, A000108.

Cf. A139702, A143426, A143435, A182969.

A:= proc(n) option remember; `if`(n=0, 1, (T->
      unapply(convert(series(1+x*T(x*T(x)), x, n+1)
      , polynom), x))(A(n-1)))
    end:
a:= n-> coeff(A(n)(x), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, May 15 2016

a[n_] := (A=x; If[n<1, 0, For[i=1, i <= n, i++, A = InverseSeries[2*(x/(1 + Sqrt[1 + 4*A + x*O[x]^n]))]]]; SeriesCoefficient[A, {x, 0, n}]); Array[a, 26] (* Jean-François Alcover, Oct 04 2016, adapted from PARI *)

{a(n)=my(A=x); if(n<1, 0, for(i=1,n,A=serreverse(2*x/(1 + sqrt(1+4*A +x*O(x^n))))); polcoeff(A, n))}

{a(n,m=1)=if(n==0,1,if(m==0,0^n,sum(k=0,n,m*binomial(n-k+m,k)/(n-k+m)*a(n-k,k))))} \\ Paul D. Hanna, Jul 09 2009

/* n-th Derivative: */
{Dx(n,F)=my(D=F);for(i=1,n,D=deriv(D));D}
/* G.f. */
{a(n)=my(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(m=0,n,Dx(m,x^(2*m+1)*A^(m+1))*A^(-2*m-2)/(m+1)!)+x*O(x^n)));polcoeff(A,n)} \\ Paul D. Hanna, Dec 18 2010

Let G(x) = x*A(x), then the following statements hold:
* G(x) = x*(1 + sqrt(1 + 4*G(G(x))))/2;
* G(x) = Series_Reversion[2*x/(1 + sqrt(1 + 4*G(x)))].
- Paul D. Hanna, May 15 2008
From Paul D. Hanna, Apr 16 2007: (Start)
G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + xB;
B = 1 + xAC;
C = 1 + xABD;
D = 1 + xABCE;
E = 1 + xABCDF ; ... (End)
From Paul D. Hanna, Jul 09 2009: (Start)
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n, then
a(n,m) = Sum_{k=0..n} m*C(n-k+m,k)/(n-k+m) * a(n-k,k) with a(0,m)=1.
(End)
G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(n+1)] *A(x)^(-2n-2)/(n+1)! ). - Paul D. Hanna, Dec 18 2010

Edited by N. J. A. Sloane, May 19 2008

A091713 G.f. satisfies A(x) = x + x*A(A(A(x))).

Original entry on oeis.org

1, 1, 3, 15, 99, 781, 7001, 69253, 742071, 8506775, 103411463, 1324477033, 17785238513, 249432247233, 3642471258187, 55246757713367, 868523130653947, 14127076257342933, 237386074703124457, 4115341407421082869, 73516094755096807279, 1351801707136238290351

Offset: 1

Paul D. Hanna, Jan 31 2004, Jun 04 2008

nonn

Conjecture: all terms are odd. - Paul D. Hanna, Dec 01 2024

			G.f.: A(x) = x + x^2 + 3*x^3 + 15*x^4 + 99*x^5 + 781*x^6 +...
From _Paul D. Hanna_, Apr 16 2007:
G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + xC;
B = A*(1 + xD);
C = B*(1 + xE);
D = C*(1 + xF);
E = D*(1 + xG) ; ...
The solution to the variables in the system of equations are
A=A(x), B=A(A(x)), C=A(A(A(x))), D=A(A(A(A(x)))), etc., where:
A(x) = x + x^2 + 3*x^3 + 15*x^4 + 99*x^5 + 781*x^6 + 7001*x^7 +...
A(A(x)) = x + 2*x^2 + 8*x^3 + 46*x^4 + 330*x^5 + 2756*x^6 +...
A(A(A(x))) = x + 3*x^2 + 15*x^3 + 99*x^4 + 781*x^5 + 7001*x^6 +...
A(A(A(A(x)))) = x + 4*x^2 + 24*x^3 + 180*x^4 + 1564*x^5 +...
A(A(A(A(A(x))))) = x + 5*x^2 + 35*x^3 + 295*x^4 + 2815*x^5 +...
ALTERNATE GENERATING METHOD.
The g.f. A(x) equals the sum of products of even iterations of A(x):
A(x) = x + x*A_2(x) + x*A_2(x)*A_4(x) + x*A_2(x)*A_4(x)*A_6(x) + x*A_2(x)*A_4(x)*A_6(x)*A_8(x) +...+ Product_{k=0..n} A_{2*k}(x) +...
where A_n(x) = A_{n-1}(A(x)) is the n-th iteration of A(x) with A_0(x)=x.

Paul D. Hanna, Table of n, a(n) for n = 1..400

Cf. A002449, A030266, A087949, A088714, A088717, A120971.

Cf. A141117.

Nest[x + x (# /. x -> # /. x -> #) &, O[x], 30][[3]] (* Vladimir Reshetnikov, Aug 08 2019 *)

{a(n)=local(A);A=x+x^2;for(i=3,n, A=x+x*subst(A,x,subst(A,x,A))+x*O(x^n)); polcoeff(A,n,x)}

/* Define the n-th iteration of F: */
{ITERATE(F,n,p)=local(G=x);for(i=1,n,G=subst(F,x,G+x*O(x^p)));G}
/* A(x) equals the sum of products of even iterations of A(x): */
{a(n)=local(A=x);for(i=1,n,A=sum(m=0,n-1,prod(k=0,m,ITERATE(A,2*k,n)+x*O(x^n))));polcoeff(A,n)}

G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+2)) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007
G.f.: A(x) = G(x)/[1 + G(G(x))] where G(x) = A(A(x)) = g.f. of A141117.
G.f.: A(x) = Series_Reversion[ x/(1 + A(A(x))) ].
G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=0..n} A_{2*k}(x), where A_n(x) denotes the n-th iteration of A(x) with A_0(x)=x. - Paul D. Hanna, Jul 21 2011

A088717 G.f. satisfies: A(x) = 1 + xA(x)^2A(x*A(x)^2).

Original entry on oeis.org

1, 1, 3, 14, 84, 596, 4785, 42349, 406287, 4176971, 45640572, 526788153, 6392402793, 81247489335, 1078331283648, 14907041720241, 214187010762831, 3192620516380376, 49287883925072010, 786925082232918304, 12976244331714379149, 220728563512663520510

Offset: 0

Paul D. Hanna, Oct 12 2003 and Mar 10 2007

			G.f.: A(x) = 1 + x + 3*x^2 + 14*x^3 + 84*x^4 + 596*x^5 + 4785*x^6 +...
G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x*A*B;
B = A*(1 + x*B*C);
C = B*(1 + x*C*D);
D = C*(1 + x*D*E);
E = D*(1 + x*E*F); ...
where B(x) = A(x)*A(x*A(x)^2), C(x) = A(x)*B(x*A(x)^2),  D(x) = A(x)*C(x*A(x)^2), ...
Expansions of a few of the functions described above begin:
B(x) = 1 + 2*x + 9*x^2 + 55*x^3 + 402*x^4 + 3328*x^5 + 30312*x^6 +...
C(x) = 1 + 3*x + 18*x^2 + 138*x^3 + 1218*x^4 + 11856*x^5 + 124467*x^6 +...
D(x) = 1 + 4*x + 30*x^2 + 278*x^3 + 2901*x^4 + 32846*x^5 + 395913*x^6 +...
ALTERNATE GENERATING METHOD.
Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy:
A = 1 + x*A^2*B,
B = 1 + x*(A*B)^2*C,
C = 1 + x*(A*B*C)^2*D,
D = 1 + x*(A*B*C*D)^2*E, etc.,
then B(x) = A(x*A(x)^2), C(x) = B(x*A(x)^2), D(x) = C(x*A(x)^2), etc.,
where A(x) = 1 + x*A(x)^2*A(x*A(x)^2) is the g.f. of this sequence.
Expansions of a few of the functions described above begin:
B(x) = 1 + x + 5*x^2 + 33*x^3 + 256*x^4 + 2223*x^5 + 21058*x^6 +...
C(x) = 1 + x + 7*x^2 + 60*x^3 + 578*x^4 + 6045*x^5 + 67421*x^6 +...
D(x) = 1 + x + 9*x^2 + 95*x^3 + 1098*x^4 + 13526*x^5 + 175176*x^6 +...

Cf. A215505, A215506, A215507.

Cf. A182224, A002449, A030266, A087949, A088714, A091713, A120971.

m = 22; A[] = 0; Do[A[x] = 1 + x A[x]^2 A[x A[x]^2] + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 07 2019 *)

{a(n)=local(A=1+x);for(i=0,n,A=1+x*A^2*subst(A,x,x*A^2+x*O(x^n)));polcoeff(A,n)}

/* a(n) = [x^n] (1+x*A(x))^(2*n+1)/(2*n+1): */
{a(n)=local(A=1+x); for(i=0, n, A=sum(m=0,n,polcoeff((1+x*A+x*O(x^m))^(2*m+1)/(2*m+1),m)*x^m)+x*O(x^n));polcoeff(A,n)}

{a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(2*n+m, k)/(2*n+m)*a(n-k, k))))}

a(n) = coefficient of x^n in (1+x*A(x))^(2*n+1)/(2*n+1) where A(x) = Sum_{n=0} a(n)*x^n.
Recurrence:
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then
a(n,m) = Sum_{k=0..n} m*C(2n+m,k)/(2n+m) * a(n-k,k). [Paul D. Hanna, Dec 16 2010]
G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n)*F(x,n+1)) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007

cp's OEIS Frontend

A128325 Rectangular table, read by antidiagonals, where the g.f.s of row n, R(x,n), satisfy: R(x,n+1) = R(G(x),n) for n>=0 and x*R(x,0) = G(x) = x + x*G(G(x)) is the g.f. of A030266.

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A128326 G.f.: A(x) = 1 + G(G(G(x))), where G(x) = x + x*G(G(x)) is the g.f. of A030266.

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A125280 Row sums of triangle A125280, which is the convolution triangle of A030266.

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A125278 Convolution triangle of A030266, which shifts left under self-COMPOSE.

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A125279 G.f.: A(x) = (1/x)*series_reversion(x^2/G(x)) where G(x) is the g.f. of A030266, which shifts left under self-COMPOSE.

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A141141 The main diagonal in the table of coefficients of iterations of G(x), where G(x) = x + x*G(G(x)) = g.f. of A030266.

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A088714 G.f. satisfies A(x) = 1 + x*A(x)^2*A(x*A(x)).

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A087949 G.f. satisfies A(x) = 1 + x*A(x*A(x)).

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A128325 Rectangular table, read by antidiagonals, where the g.f.s of row n, R(x,n), satisfy: R(x,n+1) = R(G(x),n) for n>=0 and xR(x,0) = G(x) = x + xG(G(x)) is the g.f. of A030266.

A088714 G.f. satisfies A(x) = 1 + xA(x)^2A(x*A(x)).

A087949 G.f. satisfies A(x) = 1 + xA(xA(x)).

A088717 G.f. satisfies: A(x) = 1 + xA(x)^2A(x*A(x)^2).