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 1-7 of 7 results.

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

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

Views

Author

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

Keywords

Comments

Signed version of A213591.

Examples

			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! -+...

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    {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))}
  • 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=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)}

Formula

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]

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)

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]

A143436 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^4.

Original entry on oeis.org

1, 1, 4, 26, 216, 2091, 22532, 263302, 3282572, 43184125, 594892016, 8533187394, 126911650416, 1950679300314, 30905935176876, 503694878376602, 8429969774716104, 144679270141457684, 2543281262706638148, 45745868441595695376, 841201149601799641988, 15801799739741607604585
Offset: 0

Views

Author

Paul D. Hanna, Aug 14 2008

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 216*x^4 + 2091*x^5 + 22532*x^6 +...
A(x*A(x)) = 1 + x + 5*x^2 + 38*x^3 + 356*x^4 + 3801*x^5 + 44508*x^6 +...
A(x*A(x))^4 = 1 + 4*x + 26*x^2 + 216*x^3 + 2091*x^4 + 22532*x^5 +...

Crossrefs

Cf. A143426, A143435, A143437.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^4,x,x*A));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, 4*j))); \\ Seiichi Manyama, Jun 05 2025

Formula

G.f. satisfies: x - G(x) = G(x)^2*A(x)^4 where G(x*A(x)) = x.
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,4*j). (End)

A143437 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^5.

Original entry on oeis.org

1, 1, 5, 40, 405, 4745, 61551, 862050, 12831835, 200874055, 3282575310, 55693595381, 977058059380, 17668078651755, 328497282637520, 6267311264123850, 122498870023756800, 2449635783413544555, 50061311067746399725, 1044531750427750075150, 22233430278290842445120
Offset: 0

Views

Author

Paul D. Hanna, Aug 14 2008

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 40*x^3 + 405*x^4 + 4745*x^5 + 61551*x^6 +...
A(x*A(x)) = 1 + x + 6*x^2 + 55*x^3 + 620*x^4 + 7940*x^5 + 111166*x^6 +...
A(x*A(x))^5 = 1 + 5*x + 40*x^2 + 405*x^3 + 4745*x^4 + 61551*x^5 +...

Crossrefs

Cf. A143426, A143435, A143436.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^5,x,x*A));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, 5*j))); \\ Seiichi Manyama, Jun 05 2025

Formula

G.f. satisfies: x - G(x) = G(x)^2*A(x)^5 where G(x*A(x)) = x.
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,5*j). (End)
Showing 1-7 of 7 results.

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