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.

A365731 G.f. satisfies A(x) = 1 + x^4*A(x)^5*(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 5, 11, 6, 0, 35, 120, 136, 51, 285, 1330, 2310, 1771, 3036, 14950, 35100, 40950, 47502, 175392, 503440, 791120, 927520, 2272424, 7037184, 13803405, 18643560, 33997080, 98920536, 226318196, 359255325, 578590155, 1445166360, 3584815443, 6573439928
Offset: 0

Views

Author

Seiichi Manyama, Sep 17 2023

Keywords

Links

Crossrefs

Cf. A365727, A365728, A365729, A365730.
Cf. A001002, A217358, A365725.
Cf. A215342.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(k, n-4*k)*binomial(n+k+1, k)/(n+k+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(k,n-4*k) * binomial(n+k+1,k) / (n+k+1).
G.f.: (1/x) * Series_Reversion( x*(1 - x^4*(1 + x)) ). - Seiichi Manyama, Sep 24 2024

A365725 G.f. satisfies A(x) = 1 + x^3*A(x)^4*(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 4, 9, 5, 22, 78, 91, 175, 680, 1224, 1938, 6270, 14630, 24794, 63756, 166980, 322920, 720720, 1900080, 4125888, 8803008, 22151360, 51778804, 111882100, 267682272, 645736432, 1442390092, 3346519020, 8094247798, 18657762006, 42890295734
Offset: 0

Views

Author

Seiichi Manyama, Sep 17 2023

Keywords

Links

Crossrefs

Cf. A308616, A365723, A365724, A365726.
Cf. A001002, A217358, A365731.
Cf. A054514.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(k, n-3*k)*binomial(n+k+1, k)/(n+k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(k,n-3*k) * binomial(n+k+1,k) / (n+k+1).
G.f.: (1/x) * Series_Reversion( x*(1 - x^3*(1 + x)) ). - Seiichi Manyama, Sep 24 2024

A217359 Series reversion of x+x^3+x^4.

Original entry on oeis.org

1, 0, -1, -1, 3, 7, -8, -45, 0, 264, 273, -1365, -3192, 5508, 27132, -7752, -193743, -158631, 1177209, 2417415, -5673525, -23595585, 14488110, 187050435, 104481780, -1251127512, -2178989008, 6775504088, 23824892148, -23395134188, -204487059656, -57418615353, 1471227866951
Offset: 1

Views

Author

R. J. Mathar, Oct 01 2012

Keywords

Examples

			If y= x+x^3+x^4, then x=y -y^3 -y^4 +3*y^5 +7*y^6 -8*y^7 -45*y^8 +...

Links

Crossrefs

Cf. A217358 (x-x^3-x^4).

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x+x^3+x^4,{x,0,20}],x],x]] (* Vaclav Kotesovec, Sep 10 2013 *)

Formula

D-finite with recurrence 124*n*(n-1)*(n-2)*a(n) +(n-1)*(n-2)*(7*n-88)*a(n-1) +(n-2)*(870*n^2-3465*n+3347)*a(n-2) +(1243*n^3-9870*n^2+25869*n-22490)*a(n-3) +8*(4*n-15)*(2*n-7)*(4*n-17)*a(n-4) = 0.
Recurrence (order 3): 31*(n-2)*(n-1)*n*(15*n-41)*a(n) = (n-2)*(n-1)*(90*n^2 - 381*n + 400)*a(n-1) - (n-2)*(3285*n^3 - 22119*n^2 + 48706*n - 34960)*a(n-2) - 8*(2*n-5)*(4*n-13)*(4*n-11)*(15*n-26)*a(n-3). - Vaclav Kotesovec, Sep 10 2013
Lim sup n->infinity |a(n)|^(1/n) = 16/sqrt(31) = 2.8736848... - Vaclav Kotesovec, Sep 10 2013

A365609 G.f. satisfies A(x) = 1 + x^2*A(x)^4*(1 + x*A(x)).

Original entry on oeis.org

1, 0, 1, 1, 4, 9, 27, 78, 231, 715, 2193, 6954, 21999, 70840, 228896, 746650, 2447757, 8072208, 26745627, 89002364, 297344960, 996865397, 3352918429, 11310307593, 38256171642, 129718262583, 440855654827, 1501451066767, 5123671576890, 17516503865294
Offset: 0

Views

Author

Seiichi Manyama, Sep 17 2023

Keywords

Crossrefs

Cf. A001005, A025250, A055113, A217358.
Cf. A365690.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(k, n-2*k)*binomial(n+2*k+1, k)/(n+2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(k,n-2*k) * binomial(n+2*k+1,k) / (n+2*k+1).
Showing 1-4 of 4 results.

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