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

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

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.

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.

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