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 1-4 of 4 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

A215506 G.f. satisfies: A(x) = 1 + x*A(x)^4*A(x*A(x)^4).

Original entry on oeis.org

1, 1, 5, 39, 387, 4516, 59522, 864981, 13641761, 230931591, 4162594198, 79412758356, 1595971048438, 33662065586127, 742852904575909, 17107642358401929, 410242691108675077, 10223836926997402818, 264339609703677876178, 7079673476173815446742
Offset: 0

Views

Author

Paul D. Hanna, Aug 13 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 39*x^3 + 387*x^4 + 4516*x^5 + 59522*x^6 +...
G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x*A^3*B;
B = A*(1 + x*B^3*C);
C = B*(1 + x*C^3*D);
D = C*(1 + x*D^3*E);
E = D*(1 + x*E^3*F); ...
where B(x) = A(x)*A(x*A(x)^4), C(x) = A(x)*B(x*A(x)^4),  D(x) = A(x)*C(x*A(x)^4), ...
Expansions of a few of the functions described above begin:
B(x) = 1 + 2*x + 15*x^2 + 158*x^3 + 1991*x^4 + 28283*x^5 + 439880*x^6 +...
C(x) = 1 + 3*x + 30*x^2 + 402*x^3 + 6257*x^4 + 107405*x^5 + 1983027*x^6 +...
D(x) = 1 + 4*x + 50*x^2 + 816*x^3 + 15215*x^4 + 308986*x^5 + 6679230*x^6 +...
E(x) = 1 + 5*x + 75*x^2 + 1445*x^3 + 31480*x^4 + 741075*x^5 + 18447355*x^6 +...
ALTERNATE GENERATING METHOD.
Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy:
A = 1 + x*A^4*B,
B = 1 + x*(A*B)^4*C,
C = 1 + x*(A*B*C)^4*D,
D = 1 + x*(A*B*C*D)^4*E, etc.,
then B(x) = A(x*A(x)^4), C(x) = B(x*A(x)^4), D(x) = C(x*A(x)^4), etc.,
where A(x) = 1 + x*A(x)^4*A(x*A(x)^4) is the g.f. of this sequence.
Expansions of a few of the functions described above begin:
B(x) = 1 + x + 9*x^2 + 105*x^3 + 1415*x^4 + 21089*x^5 + 340100*x^6 +...
C(x) = 1 + x + 13*x^2 + 203*x^3 + 3507*x^4 + 65018*x^5 + 1274266*x^6 +...
D(x) = 1 + x + 17*x^2 + 333*x^3 + 7047*x^4 + 157359*x^5 + 3665076*x^6 +...
E(x) = 1 + x + 21*x^2 + 495*x^3 + 12419*x^4 + 325312*x^5 + 8813506*x^6 +...

Crossrefs

Cf. A088717, A215505, A215507.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=0, n, A=1+x*A^4*subst(A, x, x*A^4+x*O(x^n))); polcoeff(A, n)}
  • PARI
    /* a(n) = [x^n] (1+x*A(x))^(4*n+1)/(4*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))^(4*m+1)/(4*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(4*n+m, k)/(4*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))^(4*n+1)/(4*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(4*n+m,k)/(4*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)^3*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.

A384680 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)*A(x*A(x)^3) ).

Original entry on oeis.org

1, 1, 3, 15, 100, 805, 7442, 76750, 866818, 10586499, 138549918, 1929878820, 28459172110, 442421488758, 7225177328165, 123586748434192, 2208493015533530, 41138303109509415, 797178212982793708, 16041390159326400966, 334654194086236031816, 7227174934846895031544
Offset: 0

Views

Author

Seiichi Manyama, Jun 06 2025

Keywords

Crossrefs

Column k=1 of A384681.
Cf. A143501, A215505, A384145.

Programs

  • Mathematica
    terms = 22; A[] = 0; Do[A[x] = 1/(1-x*A[x]*A[x*A[x]^3]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jun 07 2025 *)
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-j+k, j)/(3*n-j+k)*a(n-j, j)));

Formula

See A384681.
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