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.

A364393 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^2).

Original entry on oeis.org

1, 2, -4, 20, -120, 800, -5696, 42416, -326304, 2572992, -20685696, 168920704, -1397257472, 11682707712, -98578346496, 838369268480, -7178912946688, 61842549386240, -535575159363584, 4660216874719232, -40722264390799360, 357204260381327360
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2023

Keywords

Crossrefs

Cf. A364395, A364397, A364399.
Cf. A346626.

Programs

  • Maple
    A364393 := proc(n)
    if n = 0 then
        1;
    else
        (-1)^(n-1)*add( binomial(n,k) * binomial(n+2*k-2,n-1),k=0..n)/n ;
    end if;
end proc:
seq(A364393(n),n=0..70); # R. J. Mathar, Jul 25 2023
  • Mathematica
    m = 22;
A[_] = 1;
Do[A[x_] = 1 + x*(1 + 1/A[x]^2) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Sep 05 2023 *)
  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(n+2*k-2, n-1))/n);

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A346626.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(n+2*k-2,n-1) for n > 0.
D-finite with recurrence 2*n*(2*n-1)*a(n) +(35*n^2-64*n+24) *a(n-1) +(-35*n^2+205*n-288) *a(n-2) +2*(-43*n^2+341*n-660) *a(n-3) -4*(7*n-30)*(n-5) *a(n-4) -8*(n-5)*(n-6)*a(n-5)=0. - R. J. Mathar, Jul 25 2023
a(n) = (-1)^(n-1)*n*3F2([1-n, (n+1)/2, n/2+1], [3/2, 2], -1) for n > 1. - Stefano Spezia, Oct 21 2023

A364395 G.f. satisfies A(x) = 1 + x/A(x)*(1 + 1/A(x)^2).

Original entry on oeis.org

1, 2, -8, 60, -552, 5648, -61712, 705104, -8321696, 100658368, -1241281536, 15546987648, -197234640384, 2529169695232, -32728878054144, 426864306146560, -5605439340018176, 74050470138645504, -983432207024885760, 13122261492710033408, -175836387068096147456
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2023

Keywords

Crossrefs

Cf. A364393, A364397, A364399.
Cf. A219534.

Programs

  • Maple
    A364395 := proc(n)
    if n = 0 then
        1;
    else
    (-1)^(n-1)*add( binomial(n,k) * binomial(2*n+2*k-2,n-1),k=0..n)/n ;
    end if;
end proc:
seq(A364395(n),n=0..80); # R. J. Mathar, Jul 25 2023
a := n -> `if`(n=0, 1, (-1)^(n+1)*binomial(2*(n-1), n-1)*hypergeom([n-1/2, -n, n], [(n+1)/2, n/2], -1) / n):
seq(simplify(a(n)), n = 0..20);  # Peter Luschny, Mar 03 2024
  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = 1 + x/A[x]*(1 + 1/A[x]^2) + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)
  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(2*n+2*k-2, n-1))/n);

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A219534.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(2*n+2*k-2,n-1) for n > 0.
D-finite with recurrence 9*n*(130549*n-619680) *(3*n-1)*(3*n-2)*a(n) +6*(-15361165*n^4 +161422948*n^3 -662268162*n^2 +955427047*n -435307620)*a(n-1) +4*(-908652649*n^4 +9061174176*n^3 -32838390812*n^2 +51018866685*n -28467674946)*a(n-2) -24*(n-3)*(50425637*n^3 -426659887*n^2 +1128823867*n -890225572)*a(n-3) -16*(n-3)*(n-4) *(4607885*n -6704077)*(2*n-9)*a(n-4)=0. - R. J. Mathar, Jul 25 2023
a(n) ~ c*(-1)^(n+1)*4^n*3F2([n-1/2, -n, n], [(n+1)/2, n/2], -1)*n^(-3/2), with c = 1/(4*sqrt(Pi)). - Stefano Spezia, Oct 21 2023
a(n) = (-1)^(n+1)*binomial(2*(n-1), n-1)*hypergeom([n-1/2, -n, n], [(n+1)/2, n/2], -1) / n. - Peter Luschny, Mar 03 2024

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

A371893 G.f. A(x) satisfies A(x) = 1 + x/A(x)^2 * (1 + A(x)^4).

Original entry on oeis.org

1, 2, 0, 16, -32, 336, -1472, 10944, -63744, 441088, -2866688, 19772416, -134832128, 941381632, -6585720832, 46607831040, -331406262272, 2373110628352, -17072541007872, 123438375763968, -896088779128832, 6530356893777920, -47752086733717504
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2024

Keywords

Crossrefs

Cf. A364393, A364394, A364396, A364397, A366364.

Programs

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

Formula

a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(3*n-4*k-2,n-1) for n > 0.

A371932 G.f. A(x) satisfies A(x) = 1 + x/A(x)^2 * (1 + A(x)^5).

Original entry on oeis.org

1, 2, 2, 26, 50, 706, 1650, 24282, 62370, 940610, 2554530, 39150810, 110311762, 1709993346, 4945525650, 77314273562, 228002115650, 3587763069826, 10741365151810, 169903043416730, 514833595840370, 8177978884039490, 25025386537586610
Offset: 0

Views

Author

Seiichi Manyama, Apr 13 2024

Keywords

Crossrefs

Cf. A364393, A364394, A364396, A364397, A366364, A371893.
Cf. A349312, A366268, A371562, A371341.

Programs

  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(3*n-5*k-2, n-1))/n);

Formula

a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(3*n-5*k-2,n-1) for n > 0.
a(n) == 2 (mod 8) for n > 0. - Hugo Pfoertner, Apr 13 2024
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.