A088714 -id:A088714 - cp's OEIS frontend

A379599 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 A088714.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 13, 0, 1, 4, 12, 32, 69, 0, 1, 5, 18, 58, 173, 419, 0, 1, 6, 25, 92, 321, 1054, 2809, 0, 1, 7, 33, 135, 523, 1971, 7039, 20353, 0, 1, 8, 42, 188, 790, 3248, 13158, 50632, 157199, 0, 1, 9, 52, 252, 1134, 4976, 21740, 94194, 387613, 1281993, 0

Offset: 0

Seiichi Manyama, Feb 27 2025

			Square array begins:
  1,    1,    1,     1,     1,     1,     1, ...
  0,    1,    2,     3,     4,     5,     6, ...
  0,    3,    7,    12,    18,    25,    33, ...
  0,   13,   32,    58,    92,   135,   188, ...
  0,   69,  173,   321,   523,   790,  1134, ...
  0,  419, 1054,  1971,  3248,  4976,  7260, ...
  0, 2809, 7039, 13158, 21740, 33480, 49210, ...

Columns k=0..1 give A000007, A088714.

Cf. A379168, A379598.

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

See A088714.

A213591 G.f. A(x) satisfies A( x - A(x)^2 ) = x.

Original entry on oeis.org

1, 1, 4, 24, 178, 1512, 14152, 142705, 1528212, 17211564, 202460400, 2474708496, 31310415376, 408815254832, 5495451727376, 75907303147652, 1075685334980240, 15618612118252960, 232102241507321384, 3526880759915999016, 54755450619399484512, 867928449982022915984

Offset: 1

Paul D. Hanna, Jun 14 2012

nonn

Unsigned version of A139702.
Self-convolution is A276370.
Row sums of triangle A277295.

			G.f.: A(x) = x + x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
where A(x) = x + A(A(x))^2:
A(A(x)) = x + 2*x^2 + 10*x^3 + 69*x^4 + 568*x^5 + 5250*x^6 + 52792*x^7 +...
A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
The g.f. satisfies the series:
A(x) = x + A(x)^2 + d/dx A(x)^4/2! + d^2/dx^2 A(x)^6/3! + d^3/dx^3 A(x)^8/4! +...
Logarithmic series:
log(A(x)/x) = A(x)^2/x + [d/dx A(x)^4/x]/2! + [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! +...
Also, A(x) = x*G(A(x)^2/x) where G(x) = x/A(x/G(x)^2) is the g.f. of A212411:
G(x) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 235*x^5 + 1792*x^6 + 15261*x^7 +...
Also, A(x)^2 = x*F(A(x)) where F(x) is the g.f. of A213628:
F(x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 616*x^6 + 5072*x^7 + 46013*x^8 +...

Vaclav Kotesovec, Table of n, a(n) for n = 1..300

Cf. A220379, A139702, A212411, A138740, A213628, A213639.
Cf. A088714, A088717, A091713, A120971, A140094, A140095.
Cf. A143426, A087949, A143435, A182969.
Cf. A275765, A276360, A276361, A276362, A276363, A276370.
Cf. A277295 (triangle).

terms = 22; A[] = 0; Do[A[x] = x + A[A[x]]^2 + O[x]^(terms+1) // Normal, terms+1]; CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Jan 09 2018 *)

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

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1,21,print1(a(n),", "))

b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*b(n-j, 2*j)));
a(n) = b(n-1, 1); \\ Seiichi Manyama, Jun 05 2025

G.f. satisfies:
(1) A(x) = x + A(A(x))^2.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n)/x / n! ).
(4) A(x) = x*G(A(x)^2/x) where G(x) = 1 + x*G(1-1/G(x))^2 is the g.f. of A212411.
(5) A(x)^2 = x*F(A(x)) where F(x) = 1 - (x^2/F(x))/F(x^2/F(x)) is the g.f. of A213628.
(6) x = A(A( x-x^2 - A(x)^2 )). - Paul D. Hanna, Jul 01 2012
(7) A(x) is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = x + B^2;
B = A + C^2;
C = B + D^2;
D = C + E^2;  ...
where B = A(A(x)), C = A(A(A(x))), D = A(A(A(A(x)))), etc.
...
a(n) = Sum_{k=0..n-1} A277295(n,k).
From Seiichi Manyama, Jun 05 2025: (Start)
Let b(n,k) = [x^n] (A(x)/x)^k.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(n+j+k,j)/(n+j+k) * b(n-j,2*j).
a(n) = b(n-1,1). (End)

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

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

Original entry on oeis.org

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)

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

A139702 G.f. satisfies: x = A( x + A(x)^2 ).

Original entry on oeis.org

1, -1, 4, -24, 178, -1512, 14152, -142705, 1528212, -17211564, 202460400, -2474708496, 31310415376, -408815254832, 5495451727376, -75907303147652, 1075685334980240, -15618612118252960, 232102241507321384, -3526880759915999016

Offset: 1

Paul D. Hanna, Apr 30 2008, May 20 2008

sign

Signed version of A213591.

			G.f.: A(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 +-...
A(x)^2 = x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
where A(x + A(x)^2) = x.
Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then:
G(x) = x + x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+... and
G(G(x)) = x + 2*x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+...
so that G(x) = G(G(x)) - x^2 = g.f. of A138740.
Logarithmic series:
log(A(x)/x) = -A(x)^2/x + [d/dx A(x)^4/x]/2! - [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! -+...

Paul D. Hanna, Table of n, a(n), n=1..100.

Cf. A138740, A212411, A213591.
Cf. A088714, A088717, A091713, A120971, A140094, A140095.
Cf. A143426, A087949, A143435, A182969.

nmax = 20; sol = {a[1] -> 1}; nmin = Length[sol]+1;
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[x - A[x + A[x]^2] + O[x]^(n+1), x][[nmin;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nmin, nmax}];
a /@ Range[nmax] /. sol (* Jean-François Alcover, Nov 06 2019 *)

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

/* 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=x-x^2+x*O(x^n));for(i=1,n,
A=x*exp(sum(m=0,n,(-1)^(m+1)*Dx(m,A^(2*m+2)/x)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}

Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then G(x) = G(G(x)) - x^2 = g.f. of A138740.
G.f. satisfies: A(x) = x*G(-A(x)^2/x) where G(x) = 1 + x*G(1-1/G(x))^2 is the g.f. of A212411.
G.f.: A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = 1 - x*B^2;
B = A - x*C^2;
C = B - x*D^2;
D = C - x*E^2;
E = D - x*F^2; ...
G.f. satisfies: A(x) = x*exp( Sum_{n>=0} (-1)^(n+1)*[d^n/dx^n A(x)^(2n+2)/x]/(n+1)! ). [Paul D. Hanna, Dec 18 2010]

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

A360950 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2n) A(x)^n / n!.

Original entry on oeis.org

1, 2, 12, 108, 1240, 16932, 264740, 4631320, 89270316, 1875586380, 42610756408, 1040307155304, 27157913296228, 754950111249488, 22267948484559720, 694746226969477744, 22863695087986373968, 791675941860401322852, 28776089467457429038620, 1095679176790207081120360

Offset: 0

Paul D. Hanna, Feb 26 2023

nonn

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 108*x^3 + 1240*x^4 + 16932*x^5 + 264740*x^6 + 4631320*x^7 + 89270316*x^8 + 1875586380*x^9 + ...
where
A(x) = 1 + (d/dx x^2*A(x)) + (d^2/dx^2 x^4*A(x)^2)/2! + (d^3/dx^3 x^6*A(x)^3)/3! + (d^4/dx^4 x^8*A(x)^4)/4! + (d^5/dx^5 x^10*A(x)^5)/5! + (d^6/dx^6 x^12*A(x)^6)/6! + ... + (d^n/dx^n x^(2*n)*A(x)^n)/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^2*A(x)), which begins
B(x) = x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2822*x^6 + 37820*x^7 + 578915*x^8 + 9918924*x^9 + 187558638*x^10 + ... + A229619(n)*x^n + ...
then A(x) = B'(x) and
B(x) = x * exp( 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! + (d^5/dx^5 x^11*A(x)^6)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^n)/n! + ... ).

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A229619, A356848, A088714, A303063.

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(2*m)*A^m/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^n / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*A(x)).
(3) B(x - x^2*A(x)) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^n / n! ) is the g.f. of A229619.
(4) a(n) = (n+1) * A229619(n+1) for n >= 0.
a(n) ~ c * n! * n^alfa / LambertW(1)^n, where alfa = 3*LambertW(1) + 1/(1 + LambertW(1)) = 2.33953361459... and c = 0.1926079501120681239... - Vaclav Kotesovec, Feb 27 2023

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.

Original entry on oeis.org

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).

A360973 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^n / n!.

Original entry on oeis.org

1, 3, 30, 462, 9243, 223014, 6232239, 196780545, 6905085381, 266134485876, 11167349876424, 506653305313320, 24713399794830075, 1289888647516181583, 71744797404818298768, 4237233894492578488668, 264870390088867267319274, 17473793269024825938396135

Offset: 0

Paul D. Hanna, Feb 27 2023

nonn

			G.f.: A(x) = 1 + 3*x^2 + 30*x^4 + 462*x^6 + 9243*x^8 + 223014*x^10 + 6232239*x^12 + 196780545*x^14 + 6905085381*x^16 + ... + a(n)*x^(2*n) + ...
where
A(x) = 1 + (d/dx x^3*A(x)) + (d^2/dx^2 x^6*A(x)^2)/2! + (d^3/dx^3 x^9*A(x)^3)/3! + (d^4/dx^4 x^12*A(x)^4)/4! + (d^5/dx^5 x^15*A(x)^5)/5! + (d^6/dx^6 x^18*A(x)^6)/6! + ... + (d^n/dx^n x^(3*n)*A(x)^n)/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^3*A(x)), which begins
B(x) = x + x^3 + 6*x^5 + 66*x^7 + 1027*x^9 + 20274*x^11 + 479403*x^13 + 13118703*x^15 + 406181493*x^17 + ... + A360976(n)*x^(2*n-1) + ...
then A(x) = B'(x) and
B(x) = x * exp( x^2*A(x) + (d/dx x^5*A(x)^2)/2! + (d^2/dx^2 x^8*A(x)^3)/3! + (d^3/dx^3 x^11*A(x)^4)/4! + (d^4/dx^4 x^14*A(x)^5)/5! + (d^5/dx^5 x^17*A(x)^6)/6! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A(x)^n)/n! + ... ).

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

Cf. A360950, A360974, A360975, A360976.

Cf. A229619, A356848, A088714, A303063.

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(3*m)*A^m/m!)) +O(x^(2*n+1))); polcoeff(A, 2*n)}
for(n=0, 25, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^3*A +O(x^(2*n+3))))); polcoeff(A, 2*n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^n / n!.
(2) A(x) = d/dx Series_Reversion(x - x^3*A(x)).
(3) B(x - x^3*A(x)) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^n / n! ) is the g.f. of A360976.
(4) a(n) = (2*n+1) * A360976(n+1) for n >= 0.
a(n) ~ c * n! * n^alfa * 2^n / LambertW(1)^n, where alfa = 1.50465021094584... and c = 0.36053267424501... - Vaclav Kotesovec, Feb 28 2023
alfa = (9*LambertW(1) - 1 + 3/(1 + LambertW(1)))/4. - Vaclav Kotesovec, Mar 13 2023

cp's OEIS Frontend

A379599 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 A088714.

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PARI

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A213591 G.f. A(x) satisfies A( x - A(x)^2 ) = x.

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

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Maple

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

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

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A139702 G.f. satisfies: x = A( x + A(x)^2 ).

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

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

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

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

A360950 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2n) A(x)^n / n!.

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.

A360973 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^n / n!.