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 15 results. Next

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

Views

Author

Seiichi Manyama, Mar 01 2025

Keywords

Examples

			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, ...

Crossrefs

Columns k=0..1 give A000007, A120971, A120970(n+1).
Cf. A379598, A381603.

Programs

  • PARI
    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)));

Formula

See A120971.

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)

A088714 G.f. satisfies A(x) = 1 + x*A(x)^2*A(x*A(x)).

Original entry on oeis.org

1, 1, 3, 13, 69, 419, 2809, 20353, 157199, 1281993, 10963825, 97828031, 907177801, 8716049417, 86553001779, 886573220093, 9351927111901, 101447092428243, 1130357986741545, 12923637003161409, 151479552582252239
Offset: 0

Views

Author

Paul D. Hanna, Oct 12 2003, May 22 2008

Keywords

Comments

Equals row sums of triangle A291820.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 419*x^5 + 2809*x^6 +...
The g.f. A(x) satisfies:
x*A(x) = x + x^2*A(x) + d/dx x^4*A(x)^2/2! + d^2/dx^2 x^6*A(x)^3/3! + d^3/dx^3 x^8*A(x)^4/4! +...
The logarithm of the g.f. is given by:
log(A(x)) = 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! +...
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 + x*A*B;
B = A + x*B*C;
C = B + x*C*D;
D = C + x*D*E;
E = D + x*E*F ; ...
where variables B,C,D,E,..., are formed from successive iterations of x*A(x):
B = A(x)*A(x*A(x)), C = B*A(x*B), D = C*A(x*C), E = D*A(x*D), ...;
more explicilty,
B = 1 + 2*x + 8*x^2 + 42*x^3 + 258*x^4 + 1764*x^5 + 13070*x^6 +...,
C = 1 + 3*x + 15*x^2 + 93*x^3 + 655*x^4 + 5039*x^5 + 41453*x^6 +...,
D = 1 + 4*x + 24*x^2 + 172*x^3 + 1372*x^4 + 11796*x^5 +...,
E = 1 + 5*x + 35*x^2 + 285*x^3 + 2545*x^4 + 24255*x^5 +...,
... (End)
Related expansions:
A(x*A(x)) = 1 + x + 4*x^2 + 22*x^3 + 142*x^4 + 1016*x^5 + 7838*x^6 + 64174*x^7 + 552112*x^8 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 173*x^4 + 1054*x^5 + 7039*x^6 + 50632*x^7 + 387613*x^8 +...
d/dx x^4*A(x)^2/2! = 2*x^3 + 5*x^4 + 21*x^5 + 112*x^6 + 692*x^7 + 4743*x^8 +...
d^2/dx^2 x^6*A(x)^3/3! = 5*x^4 + 21*x^5 + 112*x^6 + 696*x^7 + 4815*x^8 +...
d^3/dx^3 x^8*A(x)^4/4! = 14*x^5 + 84*x^6 + 540*x^7 + 3795*x^8 +...
d^4/dx^4 x^10*A(x)^5/5! = 42*x^6 + 330*x^7 + 2475*x^8 + 19305*x^9 +...
...
d^(n-1)/dx^(n-1) x^(2*n)*A(x)^n/n! = A000108(n)*x^(n+1) +...

Crossrefs

Cf. A291820, A088713, A212910, A212919.
Apart from signs, same as A067145. - Philippe Deléham, Jun 18 2006
Cf. A002449, A030266, A087949, A088717, A091713, A120971.

Programs

  • Mathematica
    m = 21; A[] = 1; Do[A[x] = 1 + x A[x]^2 A[x A[x]] + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 06 2019 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, n++; A = x + O(x^2); for(i=2, n, A = x / (1 - subst(A, x, A))); polcoeff(A, n))}; /* Michael Somos, May 21 2005 */
  • PARI
    {a(n)=local(A); if(n<0, 0, A=1+x+O(x^2); for(i=1,n, A=1/(1-x*A*subst(A,x,x*A)));polcoeff(A,n))}
  • PARI
    {a(n)=local(A); if(n<0, 0, A=1+x+O(x^2);for(i=0,n, A=(1/x)*serreverse(x-x^2*A));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)=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=1, n, Dx(m-1, x^(2*m-1)*A^m/m!))+x*O(x^n))); polcoeff(A, n)}
for(n=0, 25, print1(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, May 31 2012] */
{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n,A=1+(1/x)*sum(m=1, n+1, Dx(m-1, x^(2*m)*A^m/m!))+x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies:
(1) A(x) = (1/x)*Series_Reversion(x - x^2*A(x)).
(2) A(x) = 1 + (1/x)*Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n)*A(x)^n/n!.
(3) A(x) = exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^n/n! ).
(4) A(x) = 1/(1 - x*A(x)*A(x*A(x))).
(5) A(x) = f(x*A(x)) = (1-1/f(x))/x where f(x) is the g.f. of A088713.
Given g.f. A(x), then B(x) = x*A(x) satisfies 0 = f(x, B(x), B(B(x))) where f(a0, a1, a2) = a0 - a1 + a1*a2. - Michael Somos, May 21 2005
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+k+m,k)/(n+k+m) * a(n-k,k).
(End)
a(n) = Sum_{k=0..n} A291820(n+1,k). - Paul D. Hanna, Sep 01 2017

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

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

Views

Author

Paul D. Hanna, Jan 31 2004, Jun 04 2008

Keywords

Comments

Conjecture: all terms are odd. - Paul D. Hanna, Dec 01 2024

Examples

			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.

Links

Crossrefs

Cf. A002449, A030266, A087949, A088714, A088717, A120971.
Cf. A141117.

Programs

  • Mathematica
    Nest[x + x (# /. x -> # /. x -> #) &, O[x], 30][[3]] (* Vladimir Reshetnikov, Aug 08 2019 *)
  • PARI
    {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)}
  • PARI
    /* 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)}

Formula

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

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]

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

Views

Author

Paul D. Hanna, Oct 12 2003 and Mar 10 2007

Keywords

Examples

			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 +...

Crossrefs

Cf. A215505, A215506, A215507.
Cf. A182224, A002449, A030266, A087949, A088714, A091713, A120971.

Programs

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

Formula

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

A120970 G.f. A(x) satisfies A(x/A(x)^2) = 1 + x ; thus A(x) = 1 + Series_Reversion(x/A(x)^2).

Original entry on oeis.org

1, 1, 2, 9, 60, 504, 4946, 54430, 655362, 8496454, 117311198, 1711459903, 26228829200, 420370445830, 7021029571856, 121859518887327, 2192820745899978, 40831103986939664, 785429260324068156, 15585831041632684997, 318649154587152781210, 6704504768568697046504
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2006

Keywords

Comments

From Paul D. Hanna, Nov 16 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)

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 60*x^4 + 504*x^5 + 4946*x^6 + ...
Related expansions.
A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 142*x^4 + 1164*x^5 + 11221*x^6 + ...
A(A(x)-1) = 1 + x + 4*x^2 + 26*x^3 + 218*x^4 + 2151*x^5 + 23854*x^6 + ...
A(A(x)-1)^2 = 1 + 2*x + 9*x^2 + 60*x^3 + 504*x^4 + 4946*x^5 + ...
x/A(x)^2 = x - 2*x^2 - x^3 - 10*x^4 - 73*x^5 - 662*x^6 - 6842*x^7 - ...
Series_Reversion(x/A(x)^2) = x + 2*x^2 + 9*x^3 + 60*x^4 + 504*x^5 + 4946*x^6 + ...
To illustrate the formula a(n) = [x^(n-1)] 2*A(x)^(2*n)/(2*n),
form a table of coefficients in A(x)^(2*n) as follows:
  A^2:  [(1), 2,   5,   22,   142,   1164,   11221,   121848, ...];
  A^4:  [ 1, (4), 14,   64,   397,   3116,   29002,   306468, ...];
  A^6:  [ 1,  6, (27), 134,   825,   6270,   56492,   580902, ...];
  A^8:  [ 1,  8,  44, (240), 1502,  11200,   98144,   983016, ...];
  A^10: [ 1, 10,  65,  390, (2520), 18672,  160115,  1565260, ...];
  A^12: [ 1, 12,  90,  592,  3987, (29676), 250730,  2399388, ...];
  A^14: [ 1, 14, 119,  854,  6027,  45458, (381010), 3582266, ...]; ...
in which the main diagonal forms the initial terms of this sequence:
[2/2*(1), 2/4*(4), 2/6*(27), 2/8*(240), 2/10*(2520), 2/12*(29676), ...].

Links

Crossrefs

Cf. A120971; variants: A120972, A120974, A120976, A030266, A067145, A107096.
Cf. related variants: A145347, A145348, A147664, A145349, A145350. - Paul D. Hanna, Nov 16 2008
Cf. A381602.

Programs

  • Mathematica
    terms = 21; A[] = 1; Do[A[x] = 1 + x*A[A[x] - 1]^2 + O[x]^j // Normal, {j, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 15 2018 *)
  • PARI
    {a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0); A[ #A]=-Vec(subst(Ser(A),x,x/Ser(A)^2))[ #A]);A[n+1]}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* This sequence is generated when k=2, m=0: A(x/A(x)^k) = 1 + x*A(x)^m */ {a(n,k=2,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))} \\ Paul D. Hanna, Nov 16 2008
for(n=0,30,print1(a(n),", "))
  • PARI
    b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n+k, j)/(2*n+k)*b(n-j, 2*j)));
a(n) = if(n==0, 1, b(n-1, 2)); \\ Seiichi Manyama, Jun 04 2025

Formula

G.f. satisfies: A(x) = 1 + x*A(A(x) - 1)^2.
Let B(x) be the g.f. of A120971, 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)^2),
(c) A(x) = B(x/A(x)^2),
(d) A(x) = 1 + x*B(x)^2,
(e) B(x) = 1 + x*B(x)^2*B(A(x)-1)^2,
(f) A(B(x)-1) = B(A(x)-1) = B(x*B(x)^2).
a(n) = [x^(n-1)] (1/n)*A(x)^(2n) for n>=1 with a(0)=1; i.e., a(n) equals 1/n times the coefficient of x^(n-1) in A(x)^(2n) for n>=1. [Paul D. Hanna, Nov 16 2008]
From Seiichi Manyama, Jun 04 2025: (Start)
Let b(n,k) = [x^n] B(x)^k, where B(x) is the g.f. of A120971.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(2*n+k,j)/(2*n+k) * b(n-j,2*j).
a(n) = b(n-1,2) for n > 0. (End)

A120973 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3 * A(x*A(x)^3)^3.

Original entry on oeis.org

1, 1, 6, 60, 776, 11802, 201465, 3759100, 75404151, 1608036861, 36172106112, 853346084343, 21021015647574, 538868533164995, 14336235065928966, 394957784033440194, 11246848201518516044, 330520280036501809758
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2006

Keywords

Crossrefs

Cf. A120972; variants: A110447, A120971, A120975, A120977.

Programs

  • PARI
    {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)^3))[ #G]); A=Vec(((Ser(G)-1)/x)^(1/3));A[n+1]}
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n+k, j)/(3*n+k)*a(n-j, 3*j))); \\ Seiichi Manyama, Mar 01 2025

Formula

G.f. A(x) satisfies: A(x) = G(G(x)-1), A(G(x)-1) = G(A(x)-1), A(x) = G(x*A(x)^3) and A(x/G(x)^3) = G(x), where G(x) is the g.f. of A120972 and satisfies G(x/G(x)^3) = 1 + x.
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(3*n+k,j)/(3*n+k) * a(n-j,3*j). (End)

A120975 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 * A(x*A(x)^4)^4.

Original entry on oeis.org

1, 1, 8, 108, 1888, 38798, 894308, 22517256, 609112756, 17507219813, 530495478900, 16850219461706, 558608940038072, 19263089278722726, 689119527976265884, 25519081467271687938, 976447764170903902364
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2006

Keywords

Crossrefs

Cf. A120974; variants: A110447, A120971, A120973, A120977.

Programs

  • PARI
    {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)^4))[ #G]); A=Vec(((Ser(G)-1)/x)^(1/4));A[n+1]}
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n+k, j)/(4*n+k)*a(n-j, 4*j))); \\ Seiichi Manyama, Mar 01 2025

Formula

G.f. A(x) satisfies: A(x) = G(G(x)-1), A(G(x)-1) = G(A(x)-1), A(x) = G(x*A(x)^4) and A(x/G(x)^4) = G(x), where G(x) is the g.f. of A120974 and satisfies G(x/G(x)^4) = 1 + x.
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(4*n+k,j)/(4*n+k) * a(n-j,4*j). (End)
Showing 1-10 of 15 results. Next

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