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 1-6 of 6 results.

A145167 G.f. A(x) satisfies A(x/A(x)^6) = 1/(1-x).

Original entry on oeis.org

1, 1, 7, 106, 2349, 65078, 2093770, 75175383, 2941004409, 123442051582, 5500018250128, 258162075155942, 12693904947530988, 651028563908092621, 34708995997762871047, 1918449419812267920842, 109690826250327197055475
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2008

Keywords

Crossrefs

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n),B);for(n=0,n,B=serreverse(x/A^6);A=1/(1-B));polcoeff(A,n)}

Formula

G.f. satisfies: 1 - 1/A(x) = x*A( 1 - 1/A(x) )^6.
Self-convolution square yields A145168.
Self-convolution cube yields A145169.
Self-convolution 6th power yields A145170.

A145158 G.f. A(x) satisfies A(x/A(x)^2) = 1/(1-x).

Original entry on oeis.org

1, 1, 3, 16, 121, 1143, 12570, 154551, 2072547, 29829412, 455731327, 7332989616, 123548350018, 2169987439342, 39595583375433, 748541216196285, 14628467191450947, 294984129900772611, 6128372452917891216
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2008

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 121*x^4 + 1143*x^5 +...
x/A(x)^2 = x - 2*x^2 - 3*x^3 - 18*x^4 - 150*x^5 - 1518*x^6 -...
1/A(x) = 1 - x - 2*x^2 - 11*x^3 - 88*x^4 - 869*x^5 - 9876*x^6 -...
Series_Reversion[x/A(x)^2] = x + 2*x^2 + 11*x^3 + 88*x^4 + 869*x^5 +...
		

Crossrefs

Programs

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

Formula

G.f. satisfies: 1 - 1/A(x) = x*A( 1 - 1/A(x) )^2.
Self-convolution yields A145159.

A145162 G.f. A(x) satisfies A(x/A(x)^4) = 1/(1-x).

Original entry on oeis.org

1, 1, 5, 51, 757, 14058, 303443, 7313188, 192096189, 5413972155, 161972306602, 5104569475976, 168500227127871, 5800706769824992, 207552636468976072, 7697809237540240440, 295284422299359774761, 11693774821978063710405
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2008

Keywords

Crossrefs

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n),B);for(n=0,n,B=serreverse(x/A^4);A=1/(1-B));polcoeff(A,n)}

Formula

G.f. satisfies: 1 - 1/A(x) = x*A( 1 - 1/A(x) )^4.
Self-convolution square yields A145163.
Self-convolution 4th power yields A145164.

A145165 G.f. A(x) satisfies A(x/A(x)^5) = 1/(1-x).

Original entry on oeis.org

1, 1, 6, 76, 1406, 32531, 874407, 26234503, 857727024, 30087607090, 1120358453641, 43948073274103, 1805827523343241, 77390779901965470, 3447553371343457810, 159209478315871014816, 7605143367385966288569
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2008

Keywords

Crossrefs

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n),B);for(n=0,n,B=serreverse(x/A^5);A=1/(1-B));polcoeff(A,n)}

Formula

G.f. satisfies: 1 - 1/A(x) = x*A( 1 - 1/A(x) )^5.
Self-convolution 5th power yields A145166.

A145161 G.f. A(x) satisfies A(x/A(x)) = 1/(1-x)^3.

Original entry on oeis.org

1, 3, 15, 118, 1287, 17547, 279742, 5006016, 98012094, 2064544861, 46251043536, 1092948858498, 27078465176915, 700154717998512, 18825830520559743, 524889708138232101, 15140268414007624623, 450944670503507069127
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2008

Keywords

Crossrefs

Cf. A145160.

Programs

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

Formula

Self-convolution cube of A145160.

A381595 G.f. A(x) satisfies A(x) = 1/(1 - x * A(x) * A(x*A(x)))^3.

Original entry on oeis.org

1, 3, 24, 280, 4044, 67365, 1246534, 25051422, 538836147, 12279937669, 294374405652, 7382843258466, 192917842671564, 5235276617405133, 147163222059602313, 4275948043251399950, 128196303568520249238, 3959890522003241945409, 125863828745364900374059
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2025

Keywords

Crossrefs

Column k=3 of A381594.

Programs

  • PARI
    a(n, k=3) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n+j+k, j)/(3*n+j+k)*a(n-j, 3*j)));

Formula

See A381594.
G.f.: B(x)^3, where B(x) is the g.f. of A381601.
Showing 1-6 of 6 results.