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 1-5 of 5 results.

A364394 G.f. satisfies A(x) = 1 + x/A(x)*(1 + 1/A(x)).

Original entry on oeis.org

1, 2, -6, 34, -238, 1858, -15510, 135490, -1223134, 11320066, -106830502, 1024144482, -9945711566, 97634828354, -967298498358, 9659274283650, -97119829841854, 982391779220482, -9990160542904134, 102074758837531810, -1047391288012377774, 10788532748880319298
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2023

Keywords

Crossrefs

Cf. A112478, A364396, A364398.
Cf. A027307, A087197, A108424.

Programs

  • Maple
    A364394 := proc(n)
    if n = 0 then
        1;
    else
    (-1)^(n-1)*add( binomial(n,k) * binomial(2*n+k-2,n-1),k=0..n)/n ;
    end if;
end proc:
seq(A364394(n),n=0..80); # R. J. Mathar, Jul 25 2023
  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(2*n+k-2, n-1))/n);

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A027307.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(2*n+k-2,n-1) = (-1)^(n-1) * A108424(n) for n > 0.
D-finite with recurrence n*(2*n-1)*a(n) +3*(6*n^2-10*n+3)*a(n-1) +(-46*n^2+227*n-279)*a(n-2) +2*(n-3)*(2*n-7)*a(n-3)=0. - R. J. Mathar, Jul 25 2023
a(n) ~ c*(-1)^(n-1)*4^n*2F1([-n, 2*n-1], [n], -1)*n^(-3/2), with c = 1/(4*sqrt(Pi)) = A087197/4. - Stefano Spezia, Oct 21 2023

A364399 G.f. satisfies A(x) = 1 + x/A(x)^3*(1 + 1/A(x)^2).

Original entry on oeis.org

1, 2, -16, 212, -3400, 60384, -1142960, 22598832, -461250208, 9644611008, -205537131008, 4447969973888, -97482797466624, 2159242220999936, -48260706692535552, 1087076798266594048, -24652590023639251456, 562396337623786449920
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2023

Keywords

Crossrefs

Cf. A364393, A364395, A364397.
Cf. A364398, A364400.
Cf. A363380.

Programs

  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(4*n+2*k-2, n-1))/n);

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A363380.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(4*n+2*k-2,n-1) for n > 0.
a(n) ~ c*(-1)^(n-1)*256^n*27^(-n)*3F2([-n, 2*n, 2*n-1/2], [3*n/2, (3*n+1)/2], -1)*n^(-3/2), with c = (1/8)*sqrt(3/(2*Pi)). - Stefano Spezia, Oct 21 2023

A364396 G.f. satisfies A(x) = 1 + x/A(x)^2*(1 + 1/A(x)).

Original entry on oeis.org

1, 2, -10, 86, -902, 10506, -130594, 1697006, -22774094, 313205522, -4391039930, 62522730310, -901680559574, 13143551082138, -193339856081490, 2866341942620382, -42784807130635678, 642457682754511906, -9698259831536382826, 147091417979841002294
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2023

Keywords

Crossrefs

Cf. A112478, A364394, A364398.
Cf. A144097.

Programs

  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(3*n+k-2, n-1))/n);

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A144097.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(3*n+k-2,n-1) for n > 0.
a(n) ~ c*(-1)^(n+1)*4^(-n)*27^n*n^(-3/2)*2F1([-n, 3*n-1], [2*n], -1), with c = 1/(3*sqrt(3*Pi)). - Stefano Spezia, Oct 21 2023

A364400 G.f. satisfies A(x) = 1 + x/A(x)^3*(1 + 1/A(x)^3).

Original entry on oeis.org

1, 2, -18, 270, -4902, 98538, -2110794, 47227846, -1090742094, 25806364434, -622267199554, 15236456140542, -377814588773622, 9468373002766074, -239434464005544570, 6101951612867546166, -156561081975745809566, 4040863076496835880226
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2023

Keywords

Crossrefs

Cf. A364398, A364399.
Cf. A363304.

Programs

  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(4*n+3*k-2, n-1))/n);

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A363304.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(4*n+3*k-2,n-1) for n > 0.
a(n) ~ c*(-1)^(n+1)*256^n*27^(-n)*4F3([-n, 4*n/3, (4n-1)/3, (4*n+1)/3], [n, n+1/3, n+2/3], -1)*n^(-3/2), with c = (1/8)*sqrt (3/(2*Pi)). - Stefano Spezia, Oct 21 2023

A371562 G.f. A(x) satisfies A(x) = 1 + x/A(x)^3 * (1 + A(x)^5).

Original entry on oeis.org

1, 2, -2, 30, -166, 1514, -12474, 114006, -1050830, 10005138, -96772786, 951500686, -9469982966, 95267209850, -966979784554, 9891522355270, -101866781649310, 1055294818173474, -10989809960251490, 114983445265899774, -1208092406024272710
Offset: 0

Views

Author

Seiichi Manyama, Apr 13 2024

Keywords

Crossrefs

Cf. A364395, A364396, A364398, A364399, A364400, A364407.

Programs

  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(4*n-5*k-2, n-1))/n);

Formula

a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(4*n-5*k-2,n-1) for n > 0.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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