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.

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

A364398 G.f. satisfies A(x) = 1 + x/A(x)^3*(1 + 1/A(x)).

Original entry on oeis.org

1, 2, -14, 162, -2270, 35234, -582958, 10076354, -179802046, 3287029698, -61246957902, 1158889656930, -22207636788894, 430106644358242, -8405699952109166, 165557885912786818, -3282954949273886590, 65487784219460233602, -1313225110482709157518
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2023

Keywords

Crossrefs

Cf. A112478, A364394, A364396.
Cf. A364399, A364400.
Cf. A260332.

Programs

  • Maple
    A364398 := proc(n)
    if n = 0 then
        1;
    else
        (-1)^(n-1)*add( binomial(n,k) * binomial(4*n+k-2,n-1),k=0..n)/n ;
    end if;
end proc:
seq(A364398(n),n=0..70); # R. J. Mathar, Jul 25 2023
  • Mathematica
    nmax = 18; A[] = 1; Do[A[x] = 1+x/A[x]^3*(1+1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}]; CoefficientList[A[x], x] (* Jean-François Alcover, Oct 21 2023 *)
  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(4*n+k-2, n-1))/n);

Formula

a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(4*n+k-2,n-1) for n > 0.
D-finite with recurrence 2*n*(462919*n -714364)*(4*n-3) *(2*n-1)*(4*n-1)*a(n) +(625365036*n^5 -2723245780*n^4 +4202103460*n^3 -2471353250*n^2 +81675089*n +289227120)*a(n-1) +(-484851248*n^5 +5501638270*n^4 -25122933600*n^3 +57439557800*n^2 -65490996232*n +29691239955)*a(n-2) +(2*n-5)*(652184*n -1103659)*(4*n-13) *(n-3)*(4*n-11)*a(n-3)=0. - R. J. Mathar, Jul 25 2023
a(n) ~ c*(-1)^(n-1)*256^n*27^(-n)*2F1([1-n, 4*n], [3*n], -1)*n^(-3/2), with c = sqrt(3/(32*Pi)). - Stefano Spezia, Oct 21 2023

A364397 G.f. satisfies A(x) = 1 + x/A(x)^2*(1 + 1/A(x)^2).

Original entry on oeis.org

1, 2, -12, 124, -1560, 21776, -324256, 5046096, -81086112, 1335113408, -22408067200, 381942129792, -6593494698752, 115044039049728, -2025580621035520, 35943759448886528, -642162301086308864, 11541259115333684224, -208521418711421405184
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2023

Keywords

Crossrefs

Cf. A364393, A364395, A364399.
Cf. A363311.

Programs

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

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A363311.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(3*n+2*k-2,n-1) for n > 0.
a(n) ~ c*(-1)^(n+1)*27^n*4^(-n)*3F2([-n, 3*n/2, (3n-1)/2], [n, n+1/2], -1)*n^(-3/2), 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 1-6 of 6 results.

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