A120970 -id:A120970 - cp's OEIS frontend

A120971 G.f. A(x) satisfies A(x) = 1 + xA(x)^2 A( x*A(x)^2 )^2.

1, 1, 4, 26, 218, 2151, 23854, 289555, 3783568, 52624689, 772928988, 11918181144, 192074926618, 3224153299106, 56213565222834, 1015694652332437, 18982833869517376, 366384235565593176, 7292660345274942402

Offset: 0

Paul D. Hanna, Jul 20 2006

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 218*x^4 + 2151*x^5 + 23854*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 + x*B^2;
B = A*(1 + x*C^2);
C = B*(1 + x*D^2);
D = C*(1 + x*E^2);
E = D*(1 + x*F^2); ...
The above series begin:
B(x) = 1 + 2*x + 11*x^2 + 87*x^3 + 841*x^4 + 9288*x^5 + 113166*x^6 +...
C(x) = 1 + 3*x + 21*x^2 + 198*x^3 + 2204*x^4 + 27431*x^5 + 371102*x^6 +...
D(x) = 1 + 4*x + 34*x^2 + 374*x^3 + 4747*x^4 + 66350*x^5 + 996943*x^6 +...
E(x) = 1 + 5*x + 50*x^2 + 630*x^3 + 9015*x^4 + 140510*x^5 + 2334895*x^6 +...
F(x) = 1 + 6*x + 69*x^2 + 981*x^3 + 15658*x^4 + 270016*x^5 + 4933294*x^6 +...

Cf. A120970; variants: A110447, A120973, A120975, A120977.

Cf. A002449, A087949, A088714, A088717, A091713.

m = 19; A[] = 0; Do[A[x] = 1 + x A[x]^2 A[x A[x]^2]^2 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 07 2019 *)

{a(n)=local(A,G=[1,1]);for(i=1,n,G=concat(G,0); G[ #G]=-Vec(subst(Ser(G),x,x/Ser(G)^2))[ #G]); A=Vec(((Ser(G)-1)/x)^(1/2));A[n+1]}

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n+k, j)/(2*n+k)*a(n-j, 2*j))); \\ Seiichi Manyama, Mar 01 2025

G.f. A(x) satisfies:
(1) A(x) = G(G(x)-1),
(2) A(G(x)-1) = G(A(x)-1),
(3) A(x) = G(x*A(x)^2),
(4) A(x/G(x)^2) = G(x),
where G(x) is the g.f. of A120970 and satisfies G(x/G(x)^2) = 1 + x.
G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+1)^2) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007
Let B(x) = Sum_{n>=0} a(n)*x^(2*n+1), then B( x/(1+B(x)^2) ) = x. - Paul D. Hanna, Oct 30 2013
From Seiichi Manyama, Mar 01 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(2*n+k,j)/(2*n+k) * a(n-j,2*j). (End)

A120972 G.f. A(x) satisfies A(x/A(x)^3) = 1 + x ; thus A(x) = 1 + series_reversion(x/A(x)^3).

Original entry on oeis.org

1, 1, 3, 21, 217, 2814, 42510, 718647, 13270944, 263532276, 5567092665, 124143735663, 2905528740060, 71058906460091, 1809695198254281, 47861102278428198, 1311488806252697283, 37164457324943708739, 1087356593493807164289, 32801308084353988297404

Offset: 0

Paul D. Hanna, Jul 20 2006

nonn

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.

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 217*x^4 + 2814*x^5 + 42510*x^6 +...
Related expansions.
A(x)^3 = 1 + 3*x + 12*x^2 + 82*x^3 + 813*x^4 + 10212*x^5 + 150699*x^6 +...
A(A(x)-1) = 1 + x + 6*x^2 + 60*x^3 + 776*x^4 + 11802*x^5 + 201465*x^6 +...
A(A(x)-1)^3 = 1 + 3*x + 21*x^2 + 217*x^3 + 2814*x^4 + 42510*x^5 +...
x/A(x)^3 = x - 3*x^2 - 3*x^3 - 37*x^4 - 420*x^5 - 5823*x^6 -...
Series_Reversion(x/A(x)^3) = x + 3*x^2 + 21*x^3 + 217*x^4 + 2814*x^5 + 42510*x^6 +...
To illustrate the formula a(n) = [x^(n-1)] 3*A(x)^(3*n)/(3*n),
form a table of coefficients in A(x)^(3*n) as follows:
  A^3:  [(1), 3,  12,   82,    813,   10212,   150699,   2503233, ...];
  A^6:  [ 1, (6), 33,  236,   2262,   27270,   388906,   6289080, ...];
  A^9:  [ 1,  9, (63), 489,   4671,   54684,   756012,  11904813, ...];
  A^12: [ 1, 12, 102, (868),  8445,   97260,  1310040,  20112516, ...];
  A^15: [ 1, 15, 150, 1400, (14070), 161343,  2130505,  31961175, ...];
  A^18: [ 1, 18, 207, 2112,  22113, (255060), 3324003,  48876264, ...];
  A^21: [ 1, 21, 273, 3031,  33222,  388563, (5030529), 72769014, ...]; ...
in which the main diagonal forms the initial terms of this sequence:
[3/3*(1), 3/6*(6), 3/9*(63), 3/12*(868), 3/15*(14070), 3/18*(255060), ...].

Cf. A120973; variants: A120970, A120974, A120976, A030266, A067145, A107096.

Cf. A381603.

terms = 18; A[] = 1; Do[A[x] = 1 + x*A[A[x] - 1]^3 + O[x]^j // Normal, {j, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 15 2018 *)

{a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0); A[ #A]=-Vec(subst(Ser(A),x,x/Ser(A)^3))[ #A]);A[n+1]}

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x*subst(A^3,x,A-1+x*O(x^n)));polcoeff(A,n)}

/* This sequence is generated when k=3, m=0: A(x/A(x)^k) = 1 + x*A(x)^m */
{a(n, k=3, m=0)=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))}
for(n=0,25,print1(a(n),", "))

b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n+k, j)/(3*n+k)*b(n-j, 3*j)));
a(n) = if(n==0, 1, b(n-1, 3)); \\ Seiichi Manyama, Jun 04 2025

G.f. satisfies: A(x) = 1 + x*A(A(x) - 1)^3.
a(n) = [x^(n-1)] A(x)^(3*n)/n for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in A(x)^(3*n)/n for n>=1 (see comment).
Let B(x) be the g.f. of A120973, then B(x) and g.f. A(x) are related by:
(a) B(x) = A(A(x)-1),
(b) B(x) = A(x*B(x)^3),
(c) A(x) = B(x/A(x)^3),
(d) A(x) = 1 + x*B(x)^3,
(e) B(x) = 1 + x*B(x)^3*B(A(x)-1)^3,
(f) A(B(x)-1) = B(A(x)-1) = B(x*B(x)^3).
From Seiichi Manyama, Jun 04 2025: (Start)
Let b(n,k) = [x^n] B(x)^k, where B(x) is the g.f. of A120973.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(3*n+k,j)/(3*n+k) * b(n-j,3*j).
a(n) = b(n-1,3) for n > 0. (End)

A120974 G.f. A(x) satisfies A(x/A(x)^4) = 1 + x; thus A(x) = 1 + series_reversion(x/A(x)^4).

Original entry on oeis.org

1, 1, 4, 38, 532, 9329, 190312, 4340296, 108043128, 2890318936, 82209697588, 2467155342740, 77676395612884, 2554497746708964, 87449858261161216, 3107829518797739032, 114399270654847628768, 4353537522757357068296, 171010040645759712226048

Offset: 0

Paul D. Hanna, Jul 20 2006

nonn

Robert Israel, Table of n, a(n) for n = 0..250

Cf. A120975; variants: A120970, A120972, A120976, A030266, A067145, A107096.

A:= x -> 1:
for m from 1 to 30 do
  Ap:= unapply(A(x)+c*x^m,x);
  S:= series(Ap(x/Ap(x)^4)-1-x, x, m+1);
  cs:= solve(convert(S,polynom),c);
  A:= subs(c=cs, eval(Ap));
od:
seq(coeff(A(x),x,m),m=0..30);# Robert Israel, Oct 25 2019

nmax = 17; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[ A[x/A[x]^4] - 1 - x + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)

{a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0); A[ #A]=-Vec(subst(Ser(A),x,x/Ser(A)^4))[ #A]);A[n+1]}

b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n+k, j)/(4*n+k)*b(n-j, 4*j)));
a(n) = if(n==0, 1, b(n-1, 4)); \\ Seiichi Manyama, Jun 04 2025

G.f. satisfies: A(x) = 1 + x*B(x)^4 = 1 + (1 + x*C(x)^4 )^4 where B(x) and C(x) satisfy: C(x) = B(x)*B(A(x)-1), B(x) = A(A(x)-1), B(A(x)-1) = A(B(x)-1), B(x/A(x)^4) = A(x), B(x) = A(x*B(x)^4) and B(x) is g.f. of A120975.
From Seiichi Manyama, Jun 04 2025: (Start)
Let b(n,k) = [x^n] B(x)^k, where B(x) is the g.f. of A120975.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(4*n+k,j)/(4*n+k) * b(n-j,4*j).
a(n) = b(n-1,4) for n > 0. (End)

A120976 G.f. A(x) satisfies A(x/A(x)^5) = 1 + x ; thus A(x) = 1 + series_reversion(x/A(x)^5).

Original entry on oeis.org

1, 1, 5, 60, 1060, 23430, 602001, 17281760, 541258390, 18210836060, 651246905140, 24566101401035, 971933892729980, 40156993344526860, 1726753293393763625, 77065076699967844390, 3561820706538663354320, 170162336673835615653925, 8389644485709060522744640

Offset: 0

Paul D. Hanna, Jul 20 2006

nonn

Cf. A120977; variants: A120970, A120972, A120974, A030266, A067145, A107096.

{a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0); A[ #A]=-Vec(subst(Ser(A),x,x/Ser(A)^5))[ #A]);A[n+1]}

b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n+k, j)/(5*n+k)*b(n-j, 5*j)));
a(n) = if(n==0, 1, b(n-1, 5)); \\ Seiichi Manyama, Jun 04 2025

G.f. satisfies: A(x) = 1 + x*B(x)^5 = 1 + (1 + x*C(x)^5 )^5 where B(x) and C(x) satisfy: C(x) = B(x)*B(A(x)-1), B(x) = A(A(x)-1), B(A(x)-1) = A(B(x)-1), B(x/A(x)^5) = A(x), B(x) = A(x*B(x)^5) and B(x) is g.f. of A120977.
From Seiichi Manyama, Jun 04 2025: (Start)
Let b(n,k) = [x^n] B(x)^k, where B(x) is the g.f. of A120977.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(5*n+k,j)/(5*n+k) * b(n-j,5*j).
a(n) = b(n-1,5) for n > 0. (End)

A120955 G.f. A(x) satisfies: Series_Reversion( x/A(x) ) / x = 2*A(x) - (1+x).

Original entry on oeis.org

1, 1, 1, 4, 25, 206, 2060, 23920, 314065, 4582300, 73393490, 1278859176, 24073541260, 486806278752, 10525038917720, 242318610557760, 5919811842140945, 152974724047702626, 4169576527021400852

Offset: 0

Paul D. Hanna, Jul 19 2006

nonn

The g.f. for A120956 = Series_Reversion( x/A(x) ) / x = 2*A(x) - (1+x), where A(x) is the g.f. of this sequence.

			A(x) = 1 + x + x^2 + 4*x^3 + 25*x^4 + 206*x^5 + 2060*x^6 +...
The g.f. of A120956 is:
series_reversion(x/A(x))/x = 1 + x + 2*x^2 + 8*x^3 + 50*x^4 + 412*x^5 +...
Compare terms to see that A120956(n) = 2*a(n) for n>=2.
The g.f. satisfies the series:
A(x) = 1+x + x*d/dx (A(x)-1)^2/2! + x^2*d^2/dx^2 (A(x)-1)^3/3! + x^3*d^3/dx^3 (A(x)-1)^4/4! + x^4*d^4/dx^4 (A(x)-1)^5/5! +...

Paul D. Hanna, Table of n, a(n) for n = 0..256

Cf. A120956, A120970 (variant), A249933.

a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,t); A[ #A]=subst(Vec(serreverse(x/Ser(A)))[ #A],t,0));A[n+1]
for(n=0, 25, print1(a(n), ", "))

Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D
{a(n)=local(A=1+x*O(x^n)); for(i=1, n, A=1+x+sum(m=1, n, x^m*Dx(m, (A-1+x*O(x^n))^(m+1)/(m+1)!) )); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 27 2012

G.f. A(x) satisfies:
(1) A( x*(2*A(x) - 1-x) ) = 2*A(x) - 1-x.
(2) A(x) = 2*A(x/A(x)) - 1 - x/A(x).
(3) A(x) = F(x/A(x)) and F(x) = A(x*F(x)) where F(x) = g.f. of A120956.
(4) A(x) = 1+x + Sum{n>=1} x^n * d^n/dx^n (A(x)-1)^(n+1) / (n+1)!. - Paul D. Hanna, Dec 27 2012
a(n) = A120956(n)/2 for n>=2.
a(n) = [x^n] A(x)^n / (2*n) for n>1; i.e., a(n) equals the coefficient of x^n in A(x)^n divided by 2*n.
a(n) ~ c * n^(n + 1/2 + log(2)) / (exp(n) * (log(2))^n), where c = 0.33794865962155... . - Vaclav Kotesovec, Aug 10 2014

A381568 G.f. A(x) satisfies A(x) = (1 + xA(xA(x)))^2.

Original entry on oeis.org

1, 2, 5, 22, 126, 884, 7149, 64688, 641836, 6888740, 79203860, 968503090, 12525131474, 170555767116, 2436592516874, 36409825487380, 567612675812796, 9211031425896752, 155283809480528788, 2714788300934206360, 49140787009610861896, 919625415852055598804, 17768937720619971300781

Offset: 0

Seiichi Manyama, Feb 28 2025

nonn

Column k=1 of A381567.

Cf. A087949, A120970, A143508, A381570.

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+2*k, j)/(n-j+k)*a(n-j, j)));

See A381567.

G.f.: B(x)^2, where B(x) is the g.f. of A143508.

A381602 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A120971.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 26, 0, 1, 4, 15, 60, 218, 0, 1, 5, 22, 103, 504, 2151, 0, 1, 6, 30, 156, 870, 4946, 23854, 0, 1, 7, 39, 220, 1329, 8511, 54430, 289555, 0, 1, 8, 49, 296, 1895, 12988, 93070, 655362, 3783568, 0, 1, 9, 60, 385, 2583, 18536, 141316, 1112382, 8496454, 52624689, 0

Offset: 0

Seiichi Manyama, Mar 01 2025

			Square array begins:
  1,     1,     1,     1,      1,      1,      1, ...
  0,     1,     2,     3,      4,      5,      6, ...
  0,     4,     9,    15,     22,     30,     39, ...
  0,    26,    60,   103,    156,    220,    296, ...
  0,   218,   504,   870,   1329,   1895,   2583, ...
  0,  2151,  4946,  8511,  12988,  18536,  25332, ...
  0, 23854, 54430, 93070, 141316, 200930, 273915, ...

Columns k=0..1 give A000007, A120971, A120970(n+1).

Cf. A379598, A381603.

a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n+k, j)/(2*n+k)*a(n-j, 2*j)));

See A120971.

A182314 G.f. satisfies: A(x) = 1 + x*A(A(x)^2 - 1).

Original entry on oeis.org

1, 1, 2, 13, 174, 4232, 182382, 14175046, 2045373678, 562261694364, 299983681820740, 314433086095052371, 652379184283729238186, 2691298717301069744228618, 22133007749002207321732828222, 363389633981231330655355989037627, 11920985732676951145747564507103687806

Offset: 0

Paul D. Hanna, Apr 24 2012

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 13*x^3 + 174*x^4 + 4232*x^5 + 182382*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 30*x^3 + 378*x^4 + 8864*x^5 + 374093*x^6 +...
A(A(x)^2 - 1) = 1 + 2*x + 13*x^2 + 174*x^3 + 4232*x^4 + 182382*x^5 +...

Cf. A182224, A120970.

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x*subst(A,x,A^2-1+x*O(x^n)));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A257813 G.f. satisfies: A(x,y) = 1-x + y*x + Series_Reversion( x/A(x,y)^2 ).

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 4, 5, 0, 0, 8, 38, 14, 0, 0, 16, 184, 262, 42, 0, 0, 32, 720, 2460, 1602, 132, 0, 0, 64, 2480, 16360, 25837, 9260, 429, 0, 0, 128, 7840, 87920, 268134, 237870, 52040, 1430, 0, 0, 256, 23296, 408128, 2109040, 3638386, 2023992, 288494, 4862, 0, 0, 512, 66048, 1701504, 13676128, 40049492, 43815744, 16394336, 1590638, 16796, 0

Offset: 0

Paul D. Hanna, May 10 2015

nonn

The rightmost nonzero numbers in this triangle form the Catalan numbers (A000108).

			This triangle begins:
1;
0, 1;
0, 2, 0;
0, 4, 5, 0;
0, 8, 38, 14, 0;
0, 16, 184, 262, 42, 0;
0, 32, 720, 2460, 1602, 132, 0;
0, 64, 2480, 16360, 25837, 9260, 429, 0;
0, 128, 7840, 87920, 268134, 237870, 52040, 1430, 0;
0, 256, 23296, 408128, 2109040, 3638386, 2023992, 288494, 4862, 0;
0, 512, 66048, 1701504, 13676128, 40049492, 43815744, 16394336, 1590638, 16796, 0;
0, 1024, 180480, 6531840, 76845728, 349863976, 653001202, 487491424, 128720399, 8765044, 58786, 0;
0, 2048, 478720, 23485440, 386423488, 2571281744, 7476451420, 9591548748, 5139351752, 991185638, 48412190, 208012, 0; ...
Row sums (A120970) begin:
[1, 1, 2, 9, 60, 504, 4946, 54430, 655362, 8496454, 117311198, ...],
the g.f. of which satisfies: G(x) = 1 + Series_Reversion(x/G(x)^2).
GENERATING FUNCTION.
G.f.: A(x,y) = 1 + x*y + x^2*(2*y) + x^3*(4*y + 5*y^2) +
x^4*(8*y + 38*y^2 + 14*y^3) +
x^5*(16*y + 184*y^2 + 262*y^3 + 42*y^4) +
x^6*(32*y + 720*y^2 + 2460*y^3 + 1602*y^4 + 132*y^5) +
x^7*(64*y + 2480*y^2 + 16360*y^3 + 25837*y^4 + 9260*y^5 + 429*y^6) +
x^8*(128*y + 7840*y^2 + 87920*y^3 + 268134*y^4 + 237870*y^5 + 52040*y^6 + 1430*y^7) +
x^9*(256*y + 23296*y^2 + 408128*y^3 + 2109040*y^4 + 3638386*y^5 + 2023992*y^6 + 288494*y^7 + 4862*y^8) +...
where
A(x,y) = 1-x + y*x + Series_Reversion( x/A(x,y)^2 ).
RELATED SERIES.
A(x/A(x,y)^2, y) = 1 + y*x + (-2*y^2 + 2*y)*x^2 +
(3*y^3 - 7*y^2 + 4*y)*x^3 +
(-4*y^4 + 6*y^3 - 10*y^2 + 8*y)*x^4 +
(5*y^5 - 27*y^4 - 18*y^3 + 24*y^2 + 16*y)*x^5 +
(-6*y^6 - 14*y^5 - 312*y^4 + 60*y^3 + 240*y^2 + 32*y)*x^6 +
(7*y^7 - 147*y^6 - 1745*y^5 - 1675*y^4 + 2360*y^3 + 1136*y^2 + 64*y)*x^7 +
(-8*y^8 - 348*y^7 - 10744*y^6 - 25146*y^5 + 10246*y^4 + 21616*y^3 + 4256*y^2 + 128*y)*x^8 +
(9*y^9 - 1361*y^8 - 60738*y^7 - 267656*y^6 - 84094*y^5 + 265552*y^4 + 133952*y^3 + 14080*y^2 + 256*y)*x^9 +...

Cf. A120970, A000108.

{T(n,k) = local(A=[1]);for(i=1,n, A=Vec(1 + (y-1)*x + serreverse(x/Ser(A)^2))); polcoeff(A[n+1],k,y)}
for(n=0,10,for(k=0,n, print1(T(n,k),", "));print(""))

G.f. A(x,y) satisfies: A(x/A(x,y)^2, y) = 1+x + (y-1)*x/A(x,y)^2.

cp's OEIS Frontend

A120971 G.f. A(x) satisfies A(x) = 1 + x*A(x)^2 * A( x*A(x)^2 )^2.

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A120972 G.f. A(x) satisfies A(x/A(x)^3) = 1 + x ; thus A(x) = 1 + series_reversion(x/A(x)^3).

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A120974 G.f. A(x) satisfies A(x/A(x)^4) = 1 + x; thus A(x) = 1 + series_reversion(x/A(x)^4).

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A120976 G.f. A(x) satisfies A(x/A(x)^5) = 1 + x ; thus A(x) = 1 + series_reversion(x/A(x)^5).

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A120955 G.f. A(x) satisfies: Series_Reversion( x/A(x) ) / x = 2*A(x) - (1+x).

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A381568 G.f. A(x) satisfies A(x) = (1 + x*A(x*A(x)))^2.

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A381602 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A120971.

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A182314 G.f. satisfies: A(x) = 1 + x*A(A(x)^2 - 1).

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A120971 G.f. A(x) satisfies A(x) = 1 + xA(x)^2 A( x*A(x)^2 )^2.

A381568 G.f. A(x) satisfies A(x) = (1 + xA(xA(x)))^2.