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 1-3 of 3 results.

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

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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

A215505 G.f. satisfies: A(x) = 1 + x*A(x)^3*A(x*A(x)^3).

Original entry on oeis.org

1, 1, 4, 25, 200, 1888, 20158, 237357, 3032188, 41554144, 605964370, 9345693140, 151727166822, 2583300560490, 45984983349166, 853637181574329, 16489023127843088, 330789284905928356, 6880312907650893934, 148151276593976715612, 3297947033016039111690
Offset: 0

Views

Author

Paul D. Hanna, Aug 13 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 200*x^4 + 1888*x^5 + 20158*x^6 +...
G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x*A^2*B;
B = A*(1 + x*B^2*C);
C = B*(1 + x*C^2*D);
D = C*(1 + x*D^2*E);
E = D*(1 + x*E^2*F); ...
where B(x) = A(x)*A(x*A(x)^3), C(x) = A(x)*B(x*A(x)^3),  D(x) = A(x)*C(x*A(x)^3), ...
Expansions of a few of the functions described above begin:
B(x) = 1 + 2*x + 12*x^2 + 100*x^3 + 998*x^4 + 11258*x^5 + 139398*x^6 +...
C(x) = 1 + 3*x + 24*x^2 + 253*x^3 + 3090*x^4 + 41646*x^5 + 604636*x^6 +...
D(x) = 1 + 4*x + 40*x^2 + 512*x^3 + 7452*x^4 + 118016*x^5 + 1990284*x^6 +...
E(x) = 1 + 5*x + 60*x^2 + 905*x^3 + 15340*x^4 + 280400*x^5 + 5417498*x^6 +...
ALTERNATE GENERATING METHOD.
Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy:
A = 1 + x*A^3*B,
B = 1 + x*(A*B)^3*C,
C = 1 + x*(A*B*C)^3*D,
D = 1 + x*(A*B*C*D)^3*E, etc.,
then B(x) = A(x*A(x)^3), C(x) = B(x*A(x)^3), D(x) = C(x*A(x)^3), etc.,
where A(x) = 1 + x*A(x)^3*A(x*A(x)^3) is the g.f. of this sequence.
Expansions of a few of the functions described above begin:
B(x) = 1 + x + 7*x^2 + 64*x^3 + 681*x^4 + 8058*x^5 + 103570*x^6 +...
C(x) = 1 + x + 10*x^2 + 121*x^3 + 1630*x^4 + 23678*x^5 + 364984*x^6 +...
D(x) = 1 + x + 13*x^2 + 196*x^3 + 3209*x^4 + 55660*x^5 + 1010248*x^6 +...
E(x) = 1 + x + 16*x^2 + 289*x^3 + 5580*x^4 + 112860*x^5 + 2367358*x^6 +...

Crossrefs

Cf. A088717, A215506, A215507.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=0, n, A=1+x*A^3*subst(A, x, x*A^3+x*O(x^n))); polcoeff(A, n)}
  • PARI
    /* a(n) = [x^n] (1+x*A(x))^(3*n+1)/(3*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))^(3*m+1)/(3*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(3*n+m, k)/(3*n+m)*a(n-k, k))))}
for(n=0,31,print1(a(n),", "))

Formula

a(n) = coefficient of x^n in (1+x*A(x))^(3*n+1)/(3*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*binomial(3*n+m,k)/(3*n+m) * a(n-k,k).
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*F(x,n+1)) for n>0 with F(x,0)=1.

A215507 G.f. satisfies: A(x) = 1 + x*A(x)^5*A(x*A(x)^5).

Original entry on oeis.org

1, 1, 6, 56, 661, 9141, 142522, 2448544, 45653707, 913964706, 19491269046, 440154262428, 10475920613965, 261802864005533, 6848792691398646, 187061918111607286, 5322557388634585229, 157460119081722965460, 4834825995463338092669, 153840302781842431823086
Offset: 0

Views

Author

Paul D. Hanna, Aug 13 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 56*x^3 + 661*x^4 + 9141*x^5 + 142522*x^6 +...
G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x*A^4*B;
B = A*(1 + x*B^4*C);
C = B*(1 + x*C^4*D);
D = C*(1 + x*D^4*E);
E = D*(1 + x*E^4*F); ...
where B(x) = A(x)*A(x*A(x)^5), C(x) = A(x)*B(x*A(x)^5),  D(x) = A(x)*C(x*A(x)^5), ...
Expansions of a few of the functions described above begin:
B(x) = 1 + 2*x + 18*x^2 + 229*x^3 + 3480*x^4 + 59466*x^5 +...
C(x) = 1 + 3*x + 36*x^2 + 585*x^3 + 11055*x^4 + 230211*x^5 +...
D(x) = 1 + 4*x + 60*x^2 + 1190*x^3 + 27040*x^4 + 669426*x^5 +...
E(x) = 1 + 5*x + 90*x^2 + 2110*x^3 + 56145*x^4 + 1616151*x^5 +...
ALTERNATE GENERATING METHOD.
Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy:
A = 1 + x*A^5*B,
B = 1 + x*(A*B)^5*C,
C = 1 + x*(A*B*C)^5*D,
D = 1 + x*(A*B*C*D)^5*E, etc.,
then B(x) = A(x*A(x)^5), C(x) = B(x*A(x)^5), D(x) = C(x*A(x)^5), etc.,
where A(x) = 1 + x*A(x)^5*A(x*A(x)^5) is the g.f. of this sequence.
Expansions of a few of the functions described above begin:
B(x) = 1 + x + 11*x^2 + 156*x^3 + 2541*x^4 + 45571*x^5 + 881403*x^6 +...
C(x) = 1 + x + 16*x^2 + 306*x^3 + 6446*x^4 + 145201*x^5 +...
D(x) = 1 + x + 21*x^2 + 506*x^3 + 13126*x^4 + 358281*x^5 +...
E(x) = 1 + x + 26*x^2 + 756*x^3 + 23331*x^4 + 750061*x^5 +...

Crossrefs

Cf. A088717, A215505, A215506.

Programs

  • Mathematica
    terms = 20; A[] = 0; Do[A[x] = 1 + x*A[x]^5 * A[x*A[x]^5] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Mar 31 2025 *)
  • PARI
    {a(n)=local(A=1+x); for(i=0, n, A=1+x*A^5*subst(A, x, x*A^5+x*O(x^n))); polcoeff(A, n)}
  • PARI
    /* a(n) = [x^n] (1+x*A(x))^(5*n+1)/(5*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))^(5*m+1)/(5*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(5*n+m, k)/(5*n+m)*a(n-k, k))))}
for(n=0,41,print1(a(n),", "))

Formula

a(n) = coefficient of x^n in (1+x*A(x))^(5*n+1)/(5*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*binomial(5*n+m,k)/(5*n+m) * a(n-k,k).
G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n)^4*F(x,n+1)) for n>0 with F(x,0)=1.
Showing 1-3 of 3 results.

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