A360293 -id:A360293 - cp's OEIS frontend

A360294 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-1-2k,k) binomial(2n-6k,n-3*k).

1, 2, 6, 20, 68, 240, 864, 3154, 11628, 43196, 161430, 606228, 2285780, 8647738, 32811378, 124804104, 475748330, 1817005536, 6951390372, 26634502642, 102189927918, 392559063268, 1509684132394, 5811772604124, 22394185567728, 86364110132930, 333329513935842

Offset: 0

Seiichi Manyama, Feb 01 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A360293, A360295.

CoefficientList[Series[1/Sqrt[1-4 x/(1+x^3)],{x,0,40}],x] (* Harvey P. Dale, Feb 06 2023 *)

a(n) = sum(k=0, n\3, (-1)^k*binomial(n-1-2*k, k)*binomial(2*n-6*k, n-3*k));

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x^3)))

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

n*a(n) = 2*(2*n-1)*a(n-1) - 2*(n-3)*a(n-3) + 2*(2*n-10)*a(n-4) - (n-6)*a(n-6).

A360295 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-1-3k,k) binomial(2n-8k,n-4*k).

Original entry on oeis.org

1, 2, 6, 20, 70, 250, 912, 3372, 12590, 47362, 179230, 681528, 2601896, 9966798, 38288420, 147453664, 569092438, 2200577502, 8523612766, 33064771524, 128438624798, 499525018638, 1944918241388, 7580283784548, 29571439970136, 115459524588322, 451157870454298

Offset: 0

Seiichi Manyama, Feb 01 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A360293, A360294.

a(n) = sum(k=0, n\4, (-1)^k*binomial(n-1-3*k, k)*binomial(2*n-8*k, n-4*k));

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x^4)))

G.f.: 1 / sqrt(1-4*x/(1+x^4)).

n*a(n) = 2*(2*n-1)*a(n-1) - 2*(n-4)*a(n-4) + 2*(2*n-13)*a(n-5) - (n-8)*a(n-8).

A360290 a(n) = Sum_{k=0..floor(n/2)} binomial(n-1-k,k) * binomial(2n-4k,n-2*k).

Original entry on oeis.org

1, 2, 6, 22, 82, 314, 1222, 4814, 19138, 76626, 308550, 1248230, 5069266, 20654602, 84392838, 345659166, 1418769154, 5834283298, 24031706246, 99134911542, 409495076050, 1693539077210, 7011618614342, 29058701620974, 120540377731266, 500443750830962

Offset: 0

Seiichi Manyama, Feb 01 2023

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A085362, A360291, A360292.

Cf. A360185, A360293, A383573.

[&+[Binomial(n-1-k, k) * Binomial(2*n-4*k, n-2*k): k in [0..Floor(n div 2)]]: n in [0..30]]; // Vincenzo Librandi, May 04 2025

Table[Sum[Binomial[n-1-k,k]* Binomial[2*n-4*k, n-2*k],{k,0,Floor[n/2]}],{n,0,35}] (* Vincenzo Librandi, May 04 2025 *)

a(n) = sum(k=0, n\2, binomial(n-1-k, k)*binomial(2*n-4*k, n-2*k));

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x^2)))

G.f.: 1 / sqrt(1-4*x/(1-x^2)).
n*a(n) = 2*(2*n-1)*a(n-1) + 2*(n-2)*a(n-2) - 2*(2*n-7)*a(n-3) - (n-4)*a(n-4).
a(n) ~ phi^(3*n) / (5^(1/4) * sqrt(Pi*n/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Feb 02 2023
a(n) = A383573(n) - A383573(n-2). - Seiichi Manyama, May 01 2025

A360313 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n-1-k,n-2k) binomial(2*k,k).

Original entry on oeis.org

1, 0, -2, -2, 4, 10, -4, -38, -22, 114, 188, -234, -914, -18, 3376, 3338, -9416, -21718, 14416, 96338, 39274, -328558, -471344, 795398, 2586064, -517690, -10453424, -8272658, 32186818, 63596494, -61876584, -307070174, -62655330, 1129250706, 1356328788

Offset: 0

Seiichi Manyama, Feb 03 2023

sign

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A360314, A360315.

Cf. A026585, A360293.

RecurrenceTable[{a[0]==1,a[1]==0,a[2]==-2,a[n]==1/n (2(n-1)a[n-1]-(5n-6)a[n-2]+2(2n-5)a[n-3])},a,{n,40}] (* Harvey P. Dale, Sep 20 2024 *)

a(n) = sum(k=0, n\2, (-1)^k*binomial(n-1-k, n-2*k)*binomial(2*k, k));

my(N=40, x='x+O('x^N)); Vec(1/sqrt(1+4*x^2/(1-x)))

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

n*a(n) = 2*(n-1)*a(n-1) - (5*n-6)*a(n-2) + 2*(2*n-5)*a(n-3).

cp's OEIS Frontend

A360294 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-1-2*k,k) * binomial(2*n-6*k,n-3*k).

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A360295 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-1-3*k,k) * binomial(2*n-8*k,n-4*k).

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A360290 a(n) = Sum_{k=0..floor(n/2)} binomial(n-1-k,k) * binomial(2*n-4*k,n-2*k).

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A360313 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n-1-k,n-2*k) * binomial(2*k,k).

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Mathematica

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Formula

A360294 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-1-2k,k) binomial(2n-6k,n-3*k).

A360295 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-1-3k,k) binomial(2n-8k,n-4*k).

A360290 a(n) = Sum_{k=0..floor(n/2)} binomial(n-1-k,k) * binomial(2n-4k,n-2*k).

A360313 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n-1-k,n-2k) binomial(2*k,k).