A121687 -id:A121687 - cp's OEIS frontend

A030266 Shifts left under COMPOSE transform with itself.

0, 1, 1, 2, 6, 23, 104, 531, 2982, 18109, 117545, 808764, 5862253, 44553224, 353713232, 2924697019, 25124481690, 223768976093, 2062614190733, 19646231085928, 193102738376890, 1956191484175505, 20401540100814142, 218825717967033373, 2411606083999341827

Offset: 0

Christian G. Bower

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 23*x^5 + 104*x^6 + ...
A(A(x)) = x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 531*x^6 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..220
David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.
David Callan, Lagrange Inversion Counts 3bar-5241-Avoiding Permutations, J. Int. Seq. 14 (2011) # 11.9.4.
N. J. A. Sloane, Transforms

Cf. A001028, A035049.
Cf. A110447.
Cf. A128325, A121687.

A:= proc(n) option remember;
      unapply(`if`(n=0, x,
      A(n-1)(x)+coeff(A(n-1)(A(n-1)(x)), x, n) *x^(n+1)), x)
    end:
a:= n-> coeff(A(n)(x),x,n):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 24 2012

A[0] = Identity; A[n_] := A[n] = Function[x, Evaluate[A[n-1][x]+Coefficient[A[n-1][A[n-1][x]], x, n]*x^(n+1)]]; a[n_] := Coefficient[A[n][x], x, n]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
CoefficientList[Nest[x + x (# /. x -> #) &, O[x], 30], x] (* Vladimir Reshetnikov, Aug 08 2019 *)

{a(n)=local(A=1+x);for(i=0,n,A=1+x*A*subst(A,x,x*A+x*O(x^n)));polcoeff(A,n)} \\ Paul D. Hanna, Mar 10 2007

{a(n)=local(A=sum(i=1,n-1,a(i)*x^i)+x*O(x^n));if(n==0,0,polcoeff((1+A)^n/n,n-1))} \\ Paul D. Hanna, Nov 18 2008

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

{a(n)=local(A=1+sum(i=1,n-1,a(i)*x^i+x*O(x^n)));
for(i=1,n,A=exp(sum(m=1,n,sum(k=0,n-m,binomial(m+k-1,k)*polcoeff(A^(2*m),k)*x^k)*x^m/m)+x*O(x^n)));polcoeff(A,n)} \\ Paul D. Hanna, Dec 15 2010

G.f. A(x) satisfies the functional equation: A(x)-x = x*A(A(x)). - Paul D. Hanna, Aug 04 2002
G.f.: A(x/(1+A(x))) = x. - Paul D. Hanna, Dec 04 2003
Suppose the functions A=A(x), B=B(x), C=C(x), etc., satisfy: A = 1 + xAB, B = 1 + xABC, C = 1 + xABCD, D = 1 + xABCDE, etc., then B(x)=A(x*A(x)), C(x)=B(x*A(x)), D(x)=C(x*A(x)), etc., where A(x) = 1 + x*A(x)*A(x*A(x)) and x*A(x) is the g.f. of this sequence (see table A128325). - Paul D. Hanna, Mar 10 2007
G.f. A(x) = x*F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+1)) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007
a(n) = [x^(n-1)] [1 + A(x)]^n/n for n>=1 with a(0)=0; i.e., a(n) equals the coefficient of x^(n-1) in [1+A(x)]^n/n for n >= 1. - Paul D. Hanna, Nov 18 2008
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+m,k)/(n+m) * a(n-k,k).
(End)
G.f. satisfies:
* A(x) = x*exp( Sum_{m>=0} {d^m/dx^m A(x)^(m+1)/x} * x^(m+1)/(m+1)! );
* A(x) = x*exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^m/y^m}*x^k]*x^m/m );
which are equivalent. - Paul D. Hanna, Dec 15 2010
The g.f. satisfies:
log(A(x)/x) = A(x) + {d/dx A(x)^2/x}*x^2/2! + {d^2/dx^2 A(x)^3/x}*x^3/3! + {d^3/dx^3 A(x)^4/x}*x^4/4! + ... - Paul D. Hanna, Dec 15 2010

A145345 G.f. satisfies: A(x/A(x)) = 1 + x*A(x).

Original entry on oeis.org

1, 1, 2, 7, 34, 203, 1398, 10706, 89120, 794347, 7502170, 74511150, 773864654, 8368430208, 93905460014, 1090519614152, 13077315637592, 161643281777801, 2056306418177832, 26887064722265250, 360939404438509866

Offset: 0

Paul D. Hanna, Nov 05 2008

nonn

From Paul D. Hanna, Nov 15 2008: (Start)
More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1. (End)

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 203*x^5 + 1398*x^6 + ...
A(x/A(x)) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 203*x^6 + 1398*x^7 + ...
A(x) = 1 + x*G(x)^2 where
G(x) = 1 + x + 3*x^2 + 14*x^3 + 83*x^4 + 574*x^5 + 4432*x^6 + ...
is the g.f. of A121687.
ALTERNATE GENERATING METHOD.
This sequence forms column zero in the following array.
Let A denote this sequence.
Start in row zero with this sequence, A, after prepending an initial '1', then repeat: drop the initial term and perform convolution with A and the remaining terms in a given row to obtain the next row:
[1, 1, 1, 2, 7, 34, 203, 1398, 10706, 89120, 794347, 7502170, ...];
[1, 2, 5, 18, 86, 502, 3387, 25496, 209242, 1843134, 17235671, ...];
[2, 7, 27, 128, 727, 4763, 34912, 280006, 2418537, 22240055, ...];
[7, 34, 169, 958, 6173, 44364, 349152, 2965098, 26864357, ...];
[34, 203, 1195, 7707, 54792, 425216, 3560600, 31842929, ...];
[203, 1398, 9308, 66310, 510689, 4231188, 37425922, ...];
[1398, 10706, 78414, 605401, 4987185, 43742924, 406387957, ...];
[10706, 89120, 705227, 5824356, 50853813, 469182452, ...];
[89120, 794347, 6707823, 58712463, 539651646, 5211277285, ...];
[794347, 7502170, 67008980, 617340184, 5942316416, 59827126712, ...]; ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A383563, A121687, A145350, A145349, A147664, A302703.

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

{a(n)=local(F=1+x);for(i=0,n,G=serreverse(x/(F+x*O(x^n)))/x;F=1+x*G^2);polcoeff(F,n)} \\ Paul D. Hanna, Nov 08 2008

/* This sequence is generated when k=1, m=1: A(x/A(x)^k) = 1 + x*A(x)^m */ {a(n,k=1,m=1)=local(A=sum(i=0,n-1,a(i,k,m)*x^i));if(n==0,1,polcoeff((m+k)/(m+k*n)*A^(m+k*n),n-1))} \\ Paul D. Hanna, Nov 15 2008

/* Prints terms 0..30 */
{A=[1];
for(m=1,30,
  B=A;
  for(i=1,m-1, C=Vec(Ser(A)*Ser(B)); B=vector(#C-1,n,C[n+1]) );
  A=concat(A,0);A[#A]=B[1]
);
A} \\ Paul D. Hanna, Jan 10 2016

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+1)))[m+1] - Vec(Ser(A))[m+1])/(m+1)); A[n+1]}
for(n=0, 30, print1(2^n*a(n), ", ")) \\ Vaclav Kotesovec, Jan 31 2023

G.f. satisfies: A(x) = 1 + x*G(x)^2 where G(x) = g.f. of A121687.
G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) = g.f. of A121687. - Paul D. Hanna, Nov 08 2008
a(n) = [x^(n-1)] (2/(n+1))*A(x)^(n+1) for n>=1 with a(0)=1; i.e., a(n) equals 2/(n+1) times the coefficient of x^(n-1) in A(x)^(n+1) for n>=1. - Paul D. Hanna, Nov 15 2008

A182954 G.f. satisfies: A(x) = 1 + xA(x) A( x*A(x) )^4.

Original entry on oeis.org

1, 1, 5, 39, 381, 4284, 53163, 710810, 10085621, 150326044, 2336828792, 37687170215, 628069684439, 10782885724300, 190248852445782, 3442896376032300, 63804661588968521, 1209314277690837796

Offset: 0

Paul D. Hanna, Dec 15 2010

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 39*x^3 + 381*x^4 + 4284*x^5 + ...
Related expansions:
A(x*A(x)) = 1 + x + 6*x^2 + 54*x^3 + 592*x^4 + 7331*x^5 + 98870*x^6 + ...
A(x*A(x))^4 = 1 + 4*x + 30*x^2 + 292*x^3 + 3305*x^4 + 41420*x^5 + ...
The g.f. satisfies:
log(A(x)) = A(x)^4*x + {d/dx x*A(x)^8}*x^2/2! + {d^2/dx^2 x^2*A(x)^12}*x^3/3! + {d^3/dx^3 x^3*A(x)^16}*x^4/4! + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A384265, A030266, A121687, A182953, A182955.

{a(n) = my(A = 1 + sum(i=1,n-1,a(i)*x^i+x*O(x^n)));
for(i=1,n, A = exp( sum(m=1,n, sum(k=0,n-m, binomial(m+k-1,k)*polcoef(A^(4*m),k)*x^k) * x^m/m ) + x*O(x^n))); polcoef(A,n)}

{a(n, m=1) = if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(n+m, k)/(n+m)*a(n-k, 4*k))))}

G.f. A(x) satisfies:
* A(x) = exp( Sum_{m>=0} {d^m/dx^m x^m*A(x)^(4m+4)} * x^(m+1)/(m+1)! );
* A(x) = exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^(4m)}*x^k]*x^m/m);
which are equivalent.
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(n+m,k)/(n+m) * a(n-k,4k).

A182953 G.f. satisfies: A(x) = 1 + xA(x) A( x*A(x) )^3.

Original entry on oeis.org

1, 1, 4, 25, 197, 1797, 18178, 198937, 2318858, 28487593, 366129764, 4896068759, 67843403960, 971032668429, 14319735032679, 217136949146091, 3379973833321141, 53936100582832901, 881318215466710693, 14731508761600217914

Offset: 0

Paul D. Hanna, Dec 15 2010

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 197*x^4 + 1797*x^5 +...
Related expansions:
A(x)^3 = 1 + 3*x + 15*x^2 + 100*x^3 + 801*x^4 + 7296*x^5 + 73174*x^6 +...
A(x*A(x)) = 1 + x + 5*x^2 + 37*x^3 + 333*x^4 + 3389*x^5 + 37634*x^6 +...
A(x*A(x))^3 = 1 + 3*x + 18*x^2 + 142*x^3 + 1311*x^4 + 13461*x^5 +...
The g.f. satisfies:
log(A(x)) = A(x)^3*x + {d/dx x*A(x)^6}*x^2/2! + {d^2/dx^2 x^2*A(x)^9}*x^3/3! + {d^3/dx^3 x^3*A(x)^12}*x^4/4! +...

Cf. A147664, A030266, A121687, A182954, A182955.

{a(n)=local(A=1+sum(i=1,n-1,a(i)*x^i+x*O(x^n)));
for(i=1,n,A=exp(sum(m=1,n,sum(k=0,n-m,binomial(m+k-1,k)*polcoeff(A^(3*m),k)*x^k)*x^m/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(n+m, k)/(n+m)*a(n-k, 3*k))))}

G.f. A(x) satisfies:
* A(x) = exp( Sum_{m>=0} {d^m/dx^m x^m*A(x)^(3m+3)} * x^(m+1)/(m+1)! );
* A(x) = exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^(3m)}*x^k]*x^m/m);
which are equivalent.
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(n+m,k)/(n+m) * a(n-k,3k).
Given g.f. A(x), then G(x) = 1 + x*A(x)^3 satisfies G(x/G(x)) = 1 + x*G(x)^2 and G(x) is the g.f. of A147664.

A383563 G.f. A(x) satisfies A( x*(1+x)/A(x)^2 ) = 1 + x.

Original entry on oeis.org

1, 1, 1, 3, 13, 72, 465, 3362, 26531, 224856, 2024188, 19202830, 190857879, 1978567663, 21319434418, 238109360460, 2750229390071, 32789591062124, 402891169846242, 5094855923807780, 66229610059651788, 884081025776797026, 12107164229698851942, 169954380180177899277, 2443554376412586234247

Offset: 0

Paul D. Hanna, May 26 2025

nonn

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 13*x^4 + 72*x^5 + 465*x^6 + 3362*x^7 + 26531*x^8 + 224856*x^9 + 2024188*x^10 + ...
 where A( x*(1+x)/A(x)^2 ) = 1 + x.
RELATED SERIES.
The g.f. of A145345 begins
B(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 203*x^5 + 1398*x^6 + 10706*x^7 + 89120*x^8 + 794347*x^9 + ...
 where B(x/B(x)) = 1 + x*B(x)
 also, B( x/A(x)^2 ) = 1 + x.
The g.f. of A121687 begins
C(x) = 1 + x + 3*x^2 + 14*x^3 + 83*x^4 + 574*x^5 + 4432*x^6 + 37244*x^7 + 335153*x^8 + 3194510*x^9 + ...
 where C(x) = 1/(1 - x*C(x*C(x))^2)
 also, C( (x/(1+x))/A(x)^2 ) = 1 + x.
C(x)^2 = 1 + 2*x + 7*x^2 + 34*x^3 + 203*x^4 + 1398*x^5 + 10706*x^6 + 89120*x^7 + 794347*x^8 + ...
 where B(x) = 1 + x*C(x)^2.

Paul D. Hanna, Table of n, a(n) for n = 0..470

Cf. A384265, A145345 (B(x)), A121687 (C(x)).

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoef(x - serreverse(x/Ser(A)) + x*serreverse(x/Ser(A)^2),#A) ); A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n along with B(x) = g.f. of A145345 and C(x) = g.f. of A121687 satisfies the following formulas.
(1) A( x*(1+x)/A(x)^2 ) = 1 + x.
(2.a) Series_Reversion( x/A(x) ) = x + x*Series_Reversion( x/A(x)^2 ).
(2.b) [x^n] A(x)^(n+1)/(n+1) = [x^(n-1)] A(x)^(2*n)/n for n >= 1.
(2.c) B(x) = 1 + x*C(x)^2.
(3.a) A(x) = B(x/A(x)) where B(x) = A(x*B(x)) = C(x/B(x)).
(3.b) A(x) = C(x/A(x)^2) where C(x) = A(x*C(x)^2) = B(x*C(x)).
(4.a) A(x) = A(x)^2 - x*C(x/A(x))^2.
(4.b) B( x/A(x)^2 ) = 1 + x.
(4.c) C( (x/(1+x))/A(x)^2 ) = 1 + x.
(4.d) B( (x/(1+x))/A(x)^2 ) = 1 + x*(1+x)/A(x)^2.
(4.e) A( (x/(1+x))/A(x)^2 + x^2/A(x)^4 ) = 1 + x*(1+x)/A(x)^2.
(5.a) A145345(n) = [x^n] B(x) = [x^n] A(x)^(n+1)/(n+1) for n >= 0.
(5.b) A121687(n) = [x^n] C(x) = [x^n] A(x)^(2*n+1)/(2*n+1) for n >= 0.
(5.c) A145345(n) = [x^(n-1)] C(x)^2 = [x^(n-1)] A(x)^(2*n)/n for n >= 1.

A384265 G.f. A(x) satisfies A( x/A(x)^2 + x^2 ) = 1 + x*A(x)^2.

Original entry on oeis.org

1, 1, 3, 16, 119, 1087, 11408, 132468, 1663047, 22234598, 313303201, 4618133168, 70815362628, 1124901511837, 18450127411436, 311636597558992, 5409374008212747, 96326565666389514, 1757212245656330130, 32798907320986196010, 625759879697614204041, 12192353855092076824051, 242419156191210763668352

Offset: 0

Paul D. Hanna, May 30 2025

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 119*x^4 + 1087*x^5 + 11408*x^6 + 132468*x^7 + 1663047*x^8 + 22234598*x^9 + 313303201*x^10 + ...
 where A( x/A(x)^2 + x^2 ) = 1 + x*A(x)^2.
RELATED SERIES.
The g.f. of A145347 begins
B(x) = 1 + x + 4*x^2 + 26*x^3 + 220*x^4 + 2203*x^5 + 24836*x^6 + 306104*x^7 + 4047988*x^8 + ... + A145347(n)*x^n + ...
 where B(x/B(x)) = 1 + x*B(x)^3
 also, B(x) = A( x*B(x) ).
The g.f. of A182954 begins
C(x) = 1 + x + 5*x^2 + 39*x^3 + 381*x^4 + 4284*x^5 + 53163*x^6 + 710810*x^7 + 10085621*x^8 + ... + A182954(n)*x^n + ...
 where C(x) = 1 + x*C(x) * C( x*C(x) )^4
 also, C(x) = A( x*C(x)^2 ).
C(x)^4 = 1 + 4*x + 26*x^2 + 220*x^3 + 2203*x^4 + 24836*x^5 + 306104*x^6 + ...
 where B(x) = 1 + x*C(x)^4.
RELATED TABLE.
The table of coefficients of x^k in A(x)^n begins
  n = 1: [ 1,  1,   3,   16,   119,   1087, ...];
  n = 2: [ 1, (2),  7,   38,   279,   2508, ...];
  n = 3: [ 1,  3, (12),  67,   489,   4338, ...];
  n = 4: [(1), 4,  18, (104),  759,   6664, ...];
  n = 5: [ 1,  5,  25,  150, (1100),  9586, ...];
  n = 6: [ 1, (6), 33,  206,  1524, (13218),...];
  n = 7: [ 1,  7,  42,  273,  2044,  17689, ...];
  n = 8: [ 1,  8, (52), 352,  2674,  23144, ...];
  n = 9: [ 1,  9,  63,  444,  3429,  29745, ...];
  n =10: [ 1, 10,  75, (550), 4325,  37672, ...];
  n =11: [ 1, 11,  88,  671,  5379,  47124, ...];
  n =12: [ 1, 12, 102,  808, (6609), 58320, ...];
  ...
in which the terms in parenthesis illustrate the property
(2.b) [x^n] A(x)^(n+1) = 2 * [x^(n-1)] A(x)^(2*n+2) for n >= 1.
Further, other terms along diagonals obey
(5.a) A145345(n) = [x^n] B(x) = [x^n] A(x)^(n+1)/(n+1), and
(5.b) A121687(n) = [x^n] C(x) = [x^n] A(x)^(2*n+1)/(2*n+1) for n >= 0.

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A383563, A145347 (B(x)), A182954 (C(x)).

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoef(x - serreverse(x/Ser(A)) + serreverse(x/Ser(A)^2)^2, #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n along with B(x) = g.f. of A145347 and C(x) = g.f. of A182954 satisfy the following formulas.
(1.a) A( x/A(x)^2 + x^2 ) = 1 + x*A(x)^2.
(1.b) A(x) = A(x)^2 - x*A( sqrt(x - x/A(x)) )^4.
(2.a) x = Series_Reversion( x/A(x) ) - Series_Reversion( x/A(x)^2 )^2.
(2.b) [x^n] A(x)^(n+1) = 2 * [x^(n-1)] A(x)^(2*n+2) for n >= 1.
(2.c) B(x) = 1 + x*C(x)^4 where B(x) = A( x*B(x) ) and C(x) = A( x*C(x)^2 ).
(3.a) A(x) = B( x/A(x) ) where B(x/B(x)) = 1 + x*B(x)^3.
(3.b) A(x) = C( x/A(x)^2 ) where C(x) = 1 + x*C(x) * C( x*C(x) )^4.
(4.a) A(x) = A(x)^2 - x*C( x/A(x) )^4.
(4.b) C(x/A(x)) = A( sqrt(x - x/A(x)) ) = A( x/A(x) * C(x/A(x))^2 ).
(4.c) B(x/A(x)^2) = 1 + x*A(x)^2 = A( x/A(x)^2 * B(x/A(x)^2) ).
(5.a) A145347(n) = [x^n] B(x) = [x^n] A(x)^(n+1)/(n+1) for n >= 0.
(5.b) A182954(n) = [x^n] C(x) = [x^n] A(x)^(2*n+1)/(2*n+1) for n >= 0.
(5.c) A145347(n) = [x^(n-1)] C(x)^4 = [x^(n-1)] A(x)^(2*n+2)*2/(n+1) for n >= 1.

A182955 G.f. satisfies: A(x) = 1 + xA(x) A( x*A(x) )^5.

Original entry on oeis.org

1, 1, 6, 56, 651, 8671, 126997, 1997798, 33260799, 580270730, 10534337521, 197986746949, 3837397114948, 76473239154148, 1563252546786254, 32716989219013821, 699959257347957763, 15288884723649589585

Offset: 0

Paul D. Hanna, Dec 15 2010

nonn

			G.f.: A(x) = 1 + x + 6*x^2 + 56*x^3 + 651*x^4 + 8671*x^5 +...
Related expansions:
A(x*A(x)) = 1 + x + 7*x^2 + 74*x^3 + 953*x^4 + 13846*x^5 +...
A(x*A(x))^5 = 1 + 5*x + 45*x^2 + 520*x^3 + 6950*x^4 + 102481*x^5 +...
The g.f. satisfies:
log(A(x)) = A(x)^5*x + {d/dx x*A(x)^10}*x^2/2! + {d^2/dx^2 x^2*A(x)^15}*x^3/3! + {d^3/dx^3 x^3*A(x)^20}*x^4/4! +...

Cf. A030266, A121687, A182953, A182954.

{a(n)=local(A=1+sum(i=1,n-1,a(i)*x^i+x*O(x^n)));
for(i=1,n,A=exp(sum(m=1,n,sum(k=0,n-m,binomial(m+k-1,k)*polcoeff(A^(5*m),k)*x^k)*x^m/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(n+m, k)/(n+m)*a(n-k, 5*k))))}

G.f. A(x) satisfies:
* A(x) = exp( Sum_{m>=0} {d^m/dx^m x^m*A(x)^(5m+5)} * x^(m+1)/(m+1)! );
* A(x) = exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^(5m)}*x^k]*x^m/m);
which are equivalent.
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(n+m,k)/(n+m) * a(n-k,5k).

cp's OEIS Frontend

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