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 1-4 of 4 results.

A157004 Transform of central binomial coefficients A000984 whose Hankel transform obeys a Somos-4 recurrence.

Original entry on oeis.org

1, 2, 6, 18, 58, 192, 650, 2232, 7746, 27096, 95376, 337404, 1198546, 4272308, 15273888, 54744268, 196646922, 707747988, 2551624304, 9213416524, 33313656888, 120604436624, 437112790668, 1585877246424, 5759085911154
Offset: 0

Views

Author

Paul Barry, Feb 20 2009

Keywords

Comments

Hankel transform is A157005. Image of A000984 under Riordan array (1,x(1-x^2)).
Diagonal of rational function 1/(1 - x - y + x^3*y^2). - Seiichi Manyama, Mar 23 2023

Examples

			G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 58*x^4 + 192*x^5 + 650*x^6 + 2232*x^7 + ...

Links

Crossrefs

Cf. A157003, A360219, A360266.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/Sqrt(1-4*x+4*x^3) )); // G. C. Greubel, Feb 26 2019
  • Mathematica
    CoefficientList[Series[1/Sqrt[1-4*x*(1-x^2)], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0, n, (2*m)!/m!^2 * x^(2*m)*(1-x)^m / (1-2*x+x*O(x^n))^(2*m+1)), n)} \\ Paul D. Hanna, Sep 21 2013
  • PARI
    my(x='x+O('x^30)); Vec(1/sqrt(1-4*x+4*x^3)) \\ G. C. Greubel, Feb 26 2019
  • Sage
    (1/sqrt(1-4*x+4*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

Formula

G.f.: 1/sqrt(1 - 4*x*(1 - x^2)).
a(n) = Sum_{k=0..n} (-1)^((n-k)/2)*(1+(-1)^(n-k))*C(k,floor((n-k)/2)) *A000984(k)/2.
G.f.: Sum_{n>=0} (2*n)!/n!^2 * x^(2*n) * (1-x)^n / (1-2*x)^(2*n+1). - Paul D. Hanna, Sep 21 2013
D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1) - 2*(2*n-3)*a(n-3). - Vaclav Kotesovec, Feb 13 2014
a(n) ~ (1/r)^n / (sqrt(Pi*n) * sqrt(3-8*r)), where r = 0.2695944364054... is the root of the equation 4*r*(1-r^2)=1. - Vaclav Kotesovec, Feb 13 2014
0 = a(n)*(16*a(n+1) - 32*a(n+3) + 10*a(n+4)) + a(n+1)*(-2*a(n+3)) + a(n+2)*(16*a(n+3) - 6*a(n+4)) + a(n+3)*(-2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Sep 03 2016

A360267 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * binomial(2*(n-3*k),n-3*k).

Original entry on oeis.org

1, 2, 6, 20, 72, 264, 984, 3712, 14136, 54224, 209200, 810912, 3155616, 12320512, 48239232, 189336192, 744722400, 2934759360, 11584470336, 45796087680, 181285742592, 718498695424, 2850802065152, 11322567705600, 45011437903104, 179088911779328
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2023

Keywords

Comments

Diagonal of rational function 1/(1 - (x + y + x^4*y^3)). - Seiichi Manyama, Mar 23 2023

Links

Crossrefs

Cf. A006139, A360266.
Cf. A360219, A374599.

Programs

  • Mathematica
    Table[Sum[Binomial[n-3k,k]Binomial[2(n-3k),n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* Harvey P. Dale, May 27 2024 *)
  • PARI
    a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x^3)))

Formula

G.f.: 1/sqrt(1 - 4*x*(1 + x^3)).
n*a(n) = 2*(2*n-1)*a(n-1) + 2*(2*n-4)*a(n-4).
a(n) ~ 1 / (2*sqrt((1 - 3*r)*Pi*n) * r^n), where r = 0.2463187933841190115229... is the positive real root of the equation -1 + 4*r + 4*r^4 = 0. - Vaclav Kotesovec, Mar 23 2023

A374599 Expansion of 1/sqrt(1 - 4*x - 8*x^4).

Original entry on oeis.org

1, 2, 6, 20, 74, 276, 1044, 3992, 15414, 59948, 234484, 921432, 3634916, 14386248, 57097704, 227166384, 905714150, 3617851980, 14475452484, 58004111160, 232737175404, 934969613528, 3760157234584, 15137340947280, 60994657996476, 245980435701752
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2024

Keywords

Links

Crossrefs

Cf. A360219, A360267.
Cf. A084609, A374598.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x-8*x^4))
  • PARI
    a(n) = sum(k=0, n\4, 2^k*binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k));

Formula

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

A360271 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * Catalan(n-3*k).

Original entry on oeis.org

1, 1, 2, 5, 13, 38, 117, 373, 1222, 4085, 13877, 47766, 166229, 583893, 2067414, 7371093, 26440789, 95355990, 345538389, 1257486165, 4593933398, 16841578325, 61938532181, 228454719830, 844882459989, 3132258655573, 11638656376150, 43337083401557
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2023

Keywords

Crossrefs

Cf. A157003.
Cf. A000108, A360219, A360272.

Programs

  • Maple
    A360271 := proc(n)
    add((-1)^k*binomial(n-3*k,k)*A000108(n-3*k),k=0..n/3) ;
end proc:
seq(A360271(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • PARI
    a(n) = sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/(1+(sqrt(1-4*x*(1-x^3)))))

Formula

G.f.: c(x * (1-x^3)), where c(x) is the g.f. of A000108.
a(n) ~ 2 * sqrt(1-3*r) / (sqrt(Pi) * n^(3/2) * r^n), where r = 0.2541737124933... is the smallest positive root of the equation 1 - 4*r + 4*r^4 = 0. - Vaclav Kotesovec, Feb 01 2023
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(-n-1)*a(n-3) +2*(4*n-11)*a(n-4) +4*(-n+5)*a(n-7)=0. - R. J. Mathar, Mar 12 2023
Showing 1-4 of 4 results.

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