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.

A360266 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k,k) * binomial(2*(n-2*k),n-2*k).

Original entry on oeis.org

1, 2, 6, 22, 82, 312, 1210, 4752, 18834, 75184, 301856, 1217604, 4930626, 20032052, 81615072, 333328532, 1364264250, 5594210292, 22977466864, 94517423444, 389316529512, 1605533230256, 6628467569292, 27393187077144, 113310732332274, 469101108803052
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2023

Keywords

Comments

Diagonal of rational function 1/(1 - (x + y + x^3*y^2)). - Seiichi Manyama, Mar 23 2023

Links

Crossrefs

Cf. A006139, A360267.
Cf. A157004, A374598.

Programs

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

Formula

G.f.: 1/sqrt(1 - 4*x*(1 + x^2)).
n*a(n) = 2*(2*n-1)*a(n-1) + 2*(2*n-3)*a(n-3).

A360219 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * binomial(2*(n-3*k),n-3*k).

Original entry on oeis.org

1, 2, 6, 20, 68, 240, 864, 3152, 11616, 43136, 161152, 604992, 2280416, 8624832, 32714688, 124399488, 474066560, 1810053120, 6922776576, 26517173760, 101710338048, 390603984896, 1501732753408, 5779500226560, 22263437981184, 85835254221824, 331193445626880
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. A157004, A360267, A374599.

Programs

  • Maple
    A360219 := proc(n)
    add((-1)^k*binomial(n-3*k,k)*binomial(2*(n-3*k),n-3*k),k=0..n/3) ;
end proc:
seq(A360219(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));
  • 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).

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).

A360272 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * Catalan(n-3*k).

Original entry on oeis.org

1, 1, 2, 5, 15, 46, 147, 485, 1642, 5669, 19883, 70646, 253755, 919925, 3361546, 12368661, 45786219, 170400470, 637200555, 2392962645, 9021255722, 34128098389, 129519490219, 492966689110, 1881289209003, 7197100511317, 27595769836714, 106032318322517
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2023

Keywords

Crossrefs

Cf. A052709, A125305.
Cf. A000108, A360267, A360271, A360274.

Programs

  • Maple
    A360272 := proc(n)
    add(binomial(n-3*k,k)*A000108(n-3*k),k=0..n/3) ;
end proc:
seq(A360272(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • PARI
    a(n) = sum(k=0, n\4, 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 = 0.2463187933841... is the smallest positive root of the equation1 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

A361753 a(n) = Sum_{k=0..floor(n/3)} binomial(2*(n-3*k),k) * binomial(2*(n-3*k),n-3*k).

Original entry on oeis.org

1, 2, 6, 20, 74, 276, 1044, 3994, 15426, 60008, 234764, 922716, 3640700, 14411952, 57210750, 227659704, 907853778, 3627085932, 14515139376, 58174092472, 233463067284, 938061587212, 3773298437204, 15193083455580, 61230698571372, 246978403761112
Offset: 0

Views

Author

Seiichi Manyama, Mar 23 2023

Keywords

Comments

Diagonal of rational function 1/(1 - (1 + (x*y)^3) * (x + y)).

Links

Crossrefs

Cf. A137635, A361752.
Cf. A360267.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(2*(n-3*k), k)*binomial(2*(n-3*k), n-3*k));
  • Python
    from math import comb
def A361753(n): return sum(comb(m:=(r:=n-3*k)<<1,k)*comb(m,r) for k in range(n//3+1)) # Chai Wah Wu, Mar 23 2023

Formula

G.f.: 1/sqrt(1 - 4*x*(1 + x^3)^2).
Recurrence: n*a(n) = 2*(2*n-1)*a(n-1) + 8*(n-2)*a(n-4) + 2*(2*n-7)*a(n-7). - Vaclav Kotesovec, Mar 23 2023

A361488 Diagonal of rational function 1/(1 - (x^3 + y^3 + x^4*y)).

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 6, 12, 6, 20, 60, 60, 90, 280, 420, 532, 1330, 2520, 3444, 6804, 14112, 21912, 37884, 77616, 133914, 223080, 432432, 793364, 1341912, 2471040, 4629196, 8076640, 14453010, 26960232, 48308832, 85794852, 157947816, 287413152, 512697900, 933072064
Offset: 0

Views

Author

Seiichi Manyama, Mar 22 2023

Keywords

Links

Crossrefs

Cf. A006139, A115962.
Cf. A360267, A361727.

Programs

  • Mathematica
    Table[Sum[Binomial[2*k,k] * Binomial[k,n-3*k], {k,0,n/3}], {n,0,20}] (* Vaclav Kotesovec, Mar 23 2023 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*k, k)*binomial(k, n-3*k));

Formula

G.f.: 1/sqrt(1 - 4 * x^3 * (1+x)).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(k,n-3*k).
From Vaclav Kotesovec, Mar 23 2023: (Start)
Recurrence: n*a(n) = 2*(2*n-3)*a(n-3) + 4*(n-2)*a(n-4).
a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 1.835086681639635368143322042736678753... is the positive real root of the equation d^4 - 4*d - 4 = 0 and c = 0.2982650309662120181812121016104223... is the largest real root of the equation 1 - 20*c + 132*c^2 - 364*c^3 + 364*c^4 = 0. (End)
Showing 1-6 of 6 results.

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