cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A139715 G.f. A(x) satisfies: A(x) = G(G(x)) where G(x) = x - A(x)^2 = g.f. of A139702.

Original entry on oeis.org

1, -2, 10, -69, 568, -5250, 52792, -566830, 6420640, -76095972, 938077528, -11975951312, 157808048792, -2140767942096, 29835756120952, -426490803168368, 6244476409802008, -93541594534237356, 1432261132629484052, -22397290780155132728
Offset: 1

Views

Author

Paul D. Hanna, Apr 30 2008

Keywords

Examples

			A(x) = x - 2*x^2 + 10*x^3 - 69*x^4 + 568*x^5 - 5250*x^6 + 52792*x^7 -+...
Let G(x) = x - A(x)^2 = g.f. of A139702:
G(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 + 14152*x^7 -+...
then A(x) = G(G(x)).

Crossrefs

Cf. A139702, A138740.

Programs

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

Formula

Series_Reversion(A(x)) = F(F(x)) = F(x) + x^2 where F(x) = g.f. of A138740.

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

Views

Author

Paul D. Hanna, Jun 14 2012

Keywords

Comments

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

Examples

			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 +...

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • PARI
    {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))}
  • PARI
    {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)}
  • PARI
    {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),", "))
  • PARI
    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

Formula

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 + x*A(x*A(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

Views

Author

Paul D. Hanna, Sep 16 2003

Keywords

Examples

			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).

Links

Crossrefs

Cf. A002449, A030266, A088714, A088717, A091713, A120971, A140092, A000108.
Cf. A139702, A143426, A143435, A182969.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • 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))}
  • PARI
    {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
  • PARI
    /* 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

Formula

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

Extensions

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

A138740 G.f. satisfies A(x) = A(A(x)) - x^2 with A(0)=0.

Original entry on oeis.org

1, 1, -2, 9, -56, 420, -3572, 33328, -334354, 3559310, -39838760, 465743720, -5658983108, 71191948512, -924554859776, 12365546196641, -169995491295312, 2398380272232272, -34680290150700800, 513390937937217088, -7773229533145403728
Offset: 1

Views

Author

Paul D. Hanna, Mar 26 2008, Mar 27 2008, Apr 30 2008

Keywords

Examples

			G.f.: A(x) = x + x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-..;
A(A(x)) = x + 2*x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
The g.f. satisfies:
A(x) = x + x^2*exp((x-A(x))/x + [d/dx (x-A(x))^2/x]/2! + [d^2/dx^2 (x-A(x))^3/x]/3! + [d^3/dx^3 (x-A(x))^4/x]/4! +...)^2.
Higher order iterations of A=A(x) may be expressed in terms of A and x:
A(A(x)) = A + x^2 ;
A(A(A(x))) = (A + A^2) + x^2 ;
A(A(A(A(x)))) = (A + 2*A^2) + (1 + 2*A)*x^2 + x^4 ;
A(A(A(A(A(x))))) = (A + 3*A^2 + 2*A^3 + A^4) + (1 + 4*A + 2*A^2)*x^2 + 2*x^4 ;
A(A(A(A(A(A(x)))))) = (A + 4*A^2 + 6*A^3 + 5*A^4) + (1 + 6*A + 10*A^2 + 8*A^3)*x^2 + (3 + 6*A + 8*A^2)*x^4 + (2 + 4*A)*x^6 + x^8 ;
A(A(A(A(A(A(A(x))))))) = (A + 5*A^2 + 12*A^3 + 18*A^4 + 14*A^5 + 10*A^6 + 4*A^7 + A^8) + (1 + 8*A + 24*A^2 + 40*A^3 + 30*A^4 + 16*A^5 + 4*A^6)*x^2 + (4 + 18*A + 40*A^2 + 24*A^3 + 8*A^4)*x^4 + (6 + 20*A + 8*A^2)*x^6 + 5*x^8 .
The sums of coefficients in the above expansions form A000278: [1,1,2,3,7,16,65,321,4546,107587,20773703,...].
Let G(x) = Series_Reversion(A(x)) = g.f. of A139702, then
G(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 +-...
G(x)^2 = x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
so that G(x)^2 = A(x) - x and G(x + G(x)^2) = x.

Crossrefs

Cf. A000278, A138739.
Cf. A139702, A276370.

Programs

  • PARI
    {a(n)=local(A=x+x^2);if(n<1,0, for(i=3,n,A=A-polcoeff(subst(A,x,A+x*O(x^i))-x^2,i)*x^i);polcoeff(A,n))}
  • PARI
    {a(n)=local(A=x); if(n<1, 0, for(i=1,n, A=serreverse(x + (A+x*O(x^n))^2)); polcoeff(serreverse(A), n))}
  • PARI
    /* n-th Derivative: */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
/* G.f.: [Paul D. Hanna, Mar 24 2011] */
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+x^2*exp(2*sum(m=0, n, Dx(m, (x-A)^(m+1)/x)/(m+1)!)+x*O(x^n))); polcoeff(A, n)}

Formula

G.f. satisfies: A(x) = x + x^2*exp( 2*Sum_{n>=0} [d^n/dx^n (x-A(x))^(n+1)/x]/(n+1)! ). - Paul D. Hanna, Mar 24 2011
Let G(x) = Series_Reversion(A(x)) = g.f. of A139702, then G(x)^2 = A(x) - x so that G(x + G(x)^2) = x.

Extensions

Edited by Paul D. Hanna, May 16 2010

A143426 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^2.

Original entry on oeis.org

1, 1, 2, 7, 32, 175, 1086, 7429, 54994, 435120, 3647686, 32192596, 297654824, 2872372828, 28841766844, 300592170551, 3244942353856, 36219458512421, 417365572999944, 4958429472475171, 60659660219655616, 763325035692109389, 9870492111677035538
Offset: 0

Views

Author

Paul D. Hanna, Aug 14 2008

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 32*x^4 + 175*x^5 + 1086*x^6 +...
A(x*A(x)) = 1 + x + 3*x^2 + 13*x^3 + 70*x^4 + 434*x^5 + 2986*x^6 +...
A(x*A(x))^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 175*x^4 + 1086*x^5 +...
Logarithmic series:
log(A(x)) = x + [d/dx x^3*A(x)^4]*A(x)^(-4)/2! + [d^2/dx^2 x^5*A(x)^6]*A(x)^(-6)/3! + [d^3/dx^3 x^7*A(x)^8]*A(x)^(-8)/4! +...

Links

Crossrefs

Cf. A139702, A087949, A143435, A182969.
Cf. A143500.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^2,x,x*A));polcoeff(A,n)}
  • PARI
    /* 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=0,n,Dx(m,x^(2*m+1)*A^(2*m+2))*A^(-2*m-2)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(n-j+k, j)/(n-j+k)*a(n-j, 2*j))); \\ Seiichi Manyama, Jun 05 2025

Formula

G.f. satisfies: x - G(x) = G(x)^2*A(x)^2 where G(x*A(x)) = x.
G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(2n+2)]*A(x)^(-2n-2)/(n+1)! ). [Paul D. Hanna, Dec 18 2010]
From Seiichi Manyama, Jun 05 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(n-j+k,j)/(n-j+k) * a(n-j,2*j). (End)

A143435 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^3.

Original entry on oeis.org

1, 1, 3, 15, 97, 738, 6297, 58630, 585543, 6200916, 69071103, 804470751, 9753459717, 122670681073, 1596129692136, 21437840848440, 296680980737270, 4224090724829151, 61794432127467450, 927795254532531834, 14282871462981487854, 225247807261125989496, 3636185180695164503129
Offset: 0

Views

Author

Paul D. Hanna, Aug 14 2008

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 97*x^4 + 738*x^5 + 6297*x^6 +...
A(x*A(x)) = 1 + x + 4*x^2 + 24*x^3 + 178*x^4 + 1511*x^5 + 14130*x^6 +...
A(x*A(x))^3 = 1 + 3*x + 15*x^2 + 97*x^3 + 738*x^4 + 6297*x^5 +...
Logarithmic series:
log(A(x)) = x*A(x) + [d/dx x^3*A(x)^6]*A(x)^(-4)/2! + [d^2/dx^2 x^5*A(x)^9]*A(x)^(-6)/3! + [d^3/dx^3 x^7*A(x)^12]*A(x)^(-8)/4! +...

Crossrefs

Cf. A143426, A143436, A143437.
Cf. A139702, A087949, A182969.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^3,x,x*A));polcoeff(A,n)}
  • PARI
    /* 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=0,n,Dx(m,x^(2*m+1)*A^(3*m+3))*A^(-2*m-2)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(n-j+k, j)/(n-j+k)*a(n-j, 3*j))); \\ Seiichi Manyama, Jun 05 2025

Formula

G.f. satisfies: x - G(x) = G(x)^2*A(x)^3 where G(x*A) = x.
G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(3*n+3)]*A(x)^(-2n-2)/(n+1)! ). [Paul D. Hanna, Dec 18 2010]
From Seiichi Manyama, Jun 05 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(n-j+k,j)/(n-j+k) * a(n-j,3*j). (End)

A153851 Nonzero coefficients of the g.f. that satisfies: A(x) = x + A(A(x))^3.

Original entry on oeis.org

1, 1, 6, 57, 683, 9474, 145815, 2430393, 43202448, 810629805, 15938815794, 326653743510, 6949638584208, 153009877730525, 3477623225388063, 81429702521625843, 1961136442605508341, 48513571089988199157
Offset: 1

Views

Author

Paul D. Hanna, Jan 21 2009

Keywords

Examples

			G.f.: A(x) = x + x^3 + 6*x^5 + 57*x^7 + 683*x^9 + 9474*x^11 +...
A(x - A(x)^3) = x where
A(x)^3 = x^3 + 3*x^5 + 21*x^7 + 208*x^9 + 2517*x^11 + 34851*x^13 +...
SYSTEM OF RELATED FUNCTIONS.
A = A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x^2*B^3;
B = A + x^2*C^3;
C = B + x^2*D^3;
D = C + x^2*E^3;
E = D + x^2*F^3; ...
where the functions xB, xC, xD, etc., are successive iterations of A(x):
x*A = A(x),
x*B = A(A(x)) = g.f. of A153852,
x*C = A(A(A(x))) = g.f. of A153853,
x*D = A(A(A(A(x)))) = g.f. of A153854, etc.
The nonzero coefficients of these functions begin:
A:[1, 1, 6, 57, 683, 9474, 145815, 2430393, 43202448,...];
B:[1, 2, 15, 165, 2213, 33693, 561867, 10053141, 190489374,...];
C:[1, 3, 27, 339, 5067, 84738, 1536867, 29687772, 603835479,...];
D:[1, 4, 42, 594, 9827, 179928, 3545637, 73988631, 1618178067,...];
E:[1, 5, 60, 945, 17180, 342765, 7316178, 164606166, 3866962617,...];
F:[1, 6, 81, 1407, 27918, 603879, 13907133, 336334443, 8466942393,...];
G:[1, 7, 105, 1995, 42938, 1001973, 24795645, 642380025, 17278647147,...];
H:[1, 8, 132, 2724, 63242, 1584768, 41975610, 1160887350, 33260962995,..]; ...
The main diagonal in the above table is A153850.

Links

Crossrefs

Cf. A153852, A153853, A153854, A153850; variants: A139702, A213591.

Programs

  • PARI
    {a(n)=local(A=x+x^2); for(i=0, n, A=serreverse(x-subst(A^3, x, x+x^2*O(x^(2*n))))) ; polcoeff(A, 2*n-1)}

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) satisfies:
(1) A(x) = Series_Reversion( x - A(x)^3 ).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n) / n!. - Paul D. Hanna, Sep 07 2020
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/x / n! ). - Paul D. Hanna, Sep 07 2020
(4) x = A(A( x-x^3 - A(x)^3 )). - Paul D. Hanna, Sep 07 2020

A182969 G.f. satisfies: A(x) = 1 + x*A(x)^3*A(x*A(x)).

Original entry on oeis.org

1, 1, 4, 23, 159, 1236, 10454, 94401, 899286, 8964253, 92961432, 998600238, 11075132605, 126489183013, 1484601117235, 17876874457054, 220546820252773, 2784446513061287, 35940592329823310, 473893641259375150
Offset: 0

Views

Author

Paul D. Hanna, Dec 18 2010

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 159*x^4 + 1236*x^5 +...
Related expansions:
A(x*A(x)) = 1 + x + 5*x^2 + 35*x^3 + 287*x^4 + 2592*x^5 + 25050*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 94*x^3 + 675*x^4 + 5331*x^5 + 45274*x^6 +...
Logarithmic series:
log(A(x)) = x*A(x)^2 + [d/dx x^3*A(x)^2]*A(x)^2/2! + [d^2/dx^2 x^5*A(x)^3]*A(x)^3/3! + [d^3/dx^3 x^7*A(x)^4]*A(x)^4/4! +...

Crossrefs

Cf. A139702, A143426, A087949, A143435.

Programs

  • Maxima
    T(n,m):=if n=m then 1 else m/n*sum(sum(T(n-m,i)*k/i*binomial(2*i-k-1,i-1),i,k,n-m)*binomial(n+k-1,n-1),k,1,n-m); makelist(T(n,1),n,1,10); /* Vladimir Kruchinin, May 07 2012 */
  • PARI
    /* n-th Derivative: */
{Dx(n,F)=local(D=F);for(i=1,n,D=deriv(D));D}
/* G.f.: */
{a(n)=local(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^(m+1)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(n+1)]*A(x)^(n+1)/(n+1)! ).
a(n) = T(n-1,1), where T(n,m) = (m/n)*sum(k=1..n-m, sum(i=k..n-m, T(n-m,i)*k/i*binomial(2*i-k-1,i-1))*binomial(n+k-1,n-1)), n>m, T(n,n)=1. [Vladimir Kruchinin, May 07 2012]
T(n,m) = m * sum(k=1..n-m, (T(n-m,k)*binomial(n+2*k-1,n+k-1))/(n+k)) for n>m, and T(n,n) = 1. [Vladimir Kruchinin, Aug 08 2012]

A140094 G.f. satisfies: A(x) = x/(1 - A(A(A(x)))).

Original entry on oeis.org

1, 1, 4, 25, 199, 1855, 19387, 221407, 2717782, 35455981, 487672243, 7029980797, 105732907498, 1653377947393, 26805765569863, 449568735630517, 7785116448484318, 138980739891821269, 2554369130466577138
Offset: 1

Views

Author

Paul D. Hanna, May 08 2008, May 20 2008

Keywords

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 199*x^5 + 1855*x^6 + 19387*x^7 +...
Iterations A_{n+1}(x) = A( A_{n}(x) ) are related as follows.
A_2(x) = 1 - Series_Reversion( A(x) ) / x;
A_3(x) = 1 - x / A(x);
A_4(x) = 1 - A(x) / A_2(x);
A_5(x) = 1 - A_2(x) / A_3(x);
A_6(x) = 1 - A_3(x) / A_4(x); ...
where the iterations of A(x) begin:
A_2(x) = x + 2*x^2 + 10*x^3 + 71*x^4 + 616*x^5 + 6119*x^6 + 67210*x^7 +...;
A_3(x) = x + 3*x^2 + 18*x^3 + 144*x^4 + 1365*x^5 + 14544*x^6 +...;
A_4(x) = x + 4*x^2 + 28*x^3 + 250*x^4 + 2584*x^5 + 29584*x^6 +...;
A_5(x) = x + 5*x^2 + 40*x^3 + 395*x^4 + 4435*x^5 + 54515*x^6 +...;
A_6(x) = x + 6*x^2 + 54*x^3 + 585*x^4 + 7104*x^5 + 93555*x^6 +...;
...
Iterations are also related by continued fractions:
A(x) = x/(1 - A_2(x)/(1 - A_4(x)/(1 - A_6(x)/(1 -...)))) ;
A_2(x) = A(x)/(1 - A_3(x)/(1 - A_5(x)/(1 - A_7(x)/(1 -...)))).

Links

Crossrefs

Cf. A140095, A088714.
Cf. A088717, A091713, A120971, A139702.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = Series_Reversion(x - x*A(A(x))).
(2) A(x) = x + Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(x))^n / n!.
(3) A(x) = x*exp( Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(x))^n/x / n! ).
Define A_{n} such that A_{n+1}(x) = A( A_{n}(x) ) with A_0(x) = x,
then A_{n}(x) = A_{n-1}/[1 - A_{n+2}(x)] ;
thus A_{n}(x) = 1 - A_{n-3}(x) / A_{n-2}(x).
G.f. A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = 1 + x*A*C;
B = A + x*B*D;
C = B + x*C*E;
D = C + x*D*F;
E = D + x*E*G; ...

Extensions

Name, formulas, and examples revised by Paul D. Hanna, Feb 03 2013

A140095 G.f. satisfies: A(x) = x/(1 - A(A(A(A(x))))).

Original entry on oeis.org

1, 1, 5, 41, 437, 5513, 78477, 1225865, 20644021, 370334137, 7017055933, 139562915193, 2899946191077, 62722686552841, 1408033260333581, 32729098457253417, 786224322656857941, 19486950945070339801, 497649167866430159197, 13078602790892074110937
Offset: 1

Views

Author

Paul D. Hanna, May 08 2008, May 20 2008

Keywords

Examples

			G.f.: A(x) = x + x^2 + 5*x^3 + 41*x^4 + 437*x^5 + 5513*x^6 + 78477*x^7 +...
Iterations A_{n+1}(x) = A( A_{n}(x) ) are related as follows.
A_2(x) = 1 - Series_Reversion(A_2(x)) / Series_Reversion(A(x));
A_3(x) = 1 - Series_Reversion(A(x)) / x;
A_4(x) = 1 - x / A(x);
A_5(x) = 1 - A(x) / A_2(x);
A_6(x) = 1 - A_2(x) / A_3(x);
A_7(x) = 1 - A_3(x) / A_4(x);
A_8(x) = 1 - A_4(x) / A_5(x); ...
where the iterations of A(x) begin:
A_2(x) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1220*x^5 + 16028*x^6 +...
A_3(x) = x + 3*x^2 + 21*x^3 + 207*x^4 + 2489*x^5 + 34259*x^6 +...
A_4(x) = x + 4*x^2 + 32*x^3 + 344*x^4 + 4408*x^5 + 63776*x^6 +...
A_5(x) = x + 5*x^2 + 45*x^3 + 525*x^4 + 7165*x^5 + 109125*x^6 +...
A_6(x) = x + 6*x^2 + 60*x^3 + 756*x^4 + 10972*x^5 + 175948*x^6 +...
A_7(x) = x + 7*x^2 + 77*x^3 + 1043*x^4 + 16065*x^5 + 271103*x^6 +...
A_8(x) = x + 8*x^2 + 96*x^3 + 1392*x^4 + 22704*x^5 + 402784*x^6 +...
...
Iterations are also related by continued fractions:
A(x) = x/(1 - A_3(x)/(1 - A_6(x)/(1 - A_9(x)/(1 -...)))) ;
A_2(x) = A(x)/(1 - A_4(x)/(1 - A_7(x)/(1 - A_10(x)/(1 -...)))) ;
A_3(x) = A_2(x)/(1 - A_5(x)/(1 - A_8(x)/(1 - A_11(x)/(1 -...)))) ;
A_4(x) = A_3(x)/(1 - A_6(x)/(1 - A_9(x)/(1 - A_12(x)/(1 -...)))) ; ...

Crossrefs

Cf. A140094, A088714, A088717, A091713, A120971, A139702.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = Series_Reversion(x - x*A(A(A(x)))).
(2) A(x) = x + Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(A(x)))^n / n!.
(3) A(x) = x*exp( Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(A(x)))^n/x / n! ).
Define A_{n} such that A_{n+1}(x) = A( A_{n}(x) ) with A_0(x) = x,
then A_{n}(x) = A_{n-1}/[1 - A_{n+3}(x)] ;
thus A_{n}(x) = 1 - A_{n-4}(x) / A_{n-3}(x).
G.f.: A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = 1 + x*A*D;
B = A + x*B*E;
C = B + x*C*F;
D = C + x*D*G;
E = D + x*E*H; ...
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