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 1-7 of 7 results.

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

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

A348957 G.f. A(x) satisfies A(x) = (1 + x * A(-x)) / (1 - x * A(x)).

Original entry on oeis.org

1, 2, 2, 10, 18, 98, 210, 1194, 2786, 16258, 39906, 236938, 601458, 3615330, 9399858, 57024426, 150947010, 922283522, 2475603138, 15212318730, 41290579410, 254909413218, 698230131858, 4327273358250, 11943274468770, 74260741616514, 206279837823650, 1286199407132554
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 05 2021

Keywords

Crossrefs

Cf. A006318, A086246.
Cf. A349310, A361638, A364407, A366266, A366364.
Cf. A112478, A364394, A364395, A366363.

Programs

  • Mathematica
    nmax = 27; A[] = 0; Do[A[x] = (1 + x A[-x])/(1 - x A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -(-1)^n a[n - 1] + Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 27}]
CoefficientList[y/.AsymptoticSolve[y-y^2+x(1+y^3)==0,y->1,{x,0,27}][[1]],x] (* Alexander Burstein, Nov 26 2021 *)

Formula

a(0) = 1; a(n) = -(-1)^n * a(n-1) + Sum_{k=0..n-1} a(k) * a(n-k-1).
a(n) ~ c * 3^(3*n/4) * (1 + sqrt(3))^n / (sqrt(2*Pi) * n^(3/2) * 2^(n/2)), where c = 3^(1/4) if n is even and c = (1 + sqrt(3))/sqrt(2) if n is odd. - Vaclav Kotesovec, Nov 14 2021
From Alexander Burstein, Nov 26 2021: (Start)
G.f.: A(-x) = 1/A(x).
G.f.: A(x) = 1 + x*(1+A(x)^3)/A(x). (End)
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(2*n-3*k-2,n-1) for n > 0. - Seiichi Manyama, Apr 11 2024

A371341 G.f. A(x) satisfies A(x) = 1 + x/A(x) * (1 + A(x)^5).

Original entry on oeis.org

1, 2, 6, 46, 330, 2778, 24094, 219318, 2048274, 19583410, 190497142, 1880184446, 18778814938, 189456108554, 1927852050830, 19763367194630, 203919590002210, 2116079501498722, 22069907395614182, 231222485352688590, 2432325883912444010
Offset: 0

Views

Author

Seiichi Manyama, Apr 12 2024

Keywords

Crossrefs

Cf. A112478, A348957, A364394, A364395, A366363, A371892.

Programs

  • Maple
    A371341 := proc(n)
    if n = 0 then
        1;
    else
        add(binomial(n,k)*binomial(2*n-5*k-2,n-1),k=0..n) ;
        (-1)^(n-1)*%/n ;
    end if;
end proc:
seq(A371341(n),n=0..60) ; # R. J. Mathar, Apr 22 2024
  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(2*n-5*k-2, n-1))/n);

Formula

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

A371892 G.f. A(x) satisfies A(x) = 1 + x/A(x) * (1 + A(x)^4).

Original entry on oeis.org

1, 2, 4, 24, 112, 688, 4032, 25856, 165888, 1103616, 7412480, 50699776, 350087168, 2444208128, 17198686208, 121945948160, 870026493952, 6242802761728, 45016506564608, 326071359897600, 2371312632397824, 17307835567636480, 126743329792327680
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2024

Keywords

Crossrefs

Cf. A112478, A348957, A364394, A364395, A366363.

Programs

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

Formula

a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(2*n-4*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 1-7 of 7 results.

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