A360291 -id:A360291 - cp's OEIS frontend

A360292 a(n) = Sum_{k=0..floor(n/4)} binomial(n-1-3k,k) binomial(2n-8k,n-4*k).

1, 2, 6, 20, 70, 254, 936, 3492, 13150, 49882, 190318, 729576, 2807816, 10841962, 41983588, 162973568, 633994982, 2471010742, 9646981054, 37718873700, 147676286078, 578883674722, 2271704404900, 8923807316892, 35087269756344, 138075819924306

Offset: 0

Seiichi Manyama, Feb 01 2023

nonn

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

Cf. A085362, A360290, A360291.

Cf. A360295, A383582.

a(n) = sum(k=0, n\4, 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).
a(n) = A383582(n) - A383582(n-4). - Seiichi Manyama, May 01 2025

A360309 a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2k,n-3k) * binomial(2*k,k).

Original entry on oeis.org

1, 0, 0, 2, 2, 2, 8, 14, 20, 46, 92, 158, 314, 630, 1176, 2274, 4498, 8674, 16804, 32990, 64358, 125414, 245832, 481674, 942912, 1850122, 3633220, 7133730, 14020694, 27578954, 54261912, 106819006, 210411028, 414619486, 817344908, 1611978734, 3180333830, 6276743430

Offset: 0

Seiichi Manyama, Feb 03 2023

nonn

Cf. A026585, A360310.

Cf. A360291, A360314.

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

my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x^3/(1-x)))

G.f.: 1 / sqrt(1-4*x^3/(1-x)).
n*a(n) = 2*(n-1)*a(n-1) - (n-2)*a(n-2) + 2*(2*n-3)*a(n-3) - 2*(2*n-6)*a(n-4).
a(n) ~ 2^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 18 2023

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

A383581 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2k,k) binomial(2(n-3k),n-3*k).

Original entry on oeis.org

1, 2, 6, 21, 74, 270, 1005, 3788, 14418, 55289, 213270, 826614, 3216629, 12558928, 49175136, 193023965, 759299438, 2992534344, 11813985377, 46709675040, 184928644350, 733047010709, 2908981549006, 11555513379450, 45945148281437, 182835149061920, 728149606630164

Offset: 0

Seiichi Manyama, Apr 30 2025

nonn

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

Cf. A026375, A383573, A383582.
Cf. A376791, A383571.
Cf. A360291.

[&+[Binomial(n-2*k,k) * Binomial(2*(n-3*k),n-3*k): k in [0..n div 3]]: n in [0..25]]; // Vincenzo Librandi, May 02 2025

Table[Sum[Binomial[n-2*k,k]* Binomial[2*(n-3*k),n-3*k],{k,0,Floor[n/3]}],{n,0,30}] (* Vincenzo Librandi, May 02 2025 *)

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

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

cp's OEIS Frontend

A360292 a(n) = Sum_{k=0..floor(n/4)} binomial(n-1-3*k,k) * binomial(2*n-8*k,n-4*k).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A360309 a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2*k,n-3*k) * binomial(2*k,k).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A383581 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k,k) * binomial(2*(n-3*k),n-3*k).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A360292 a(n) = Sum_{k=0..floor(n/4)} binomial(n-1-3k,k) binomial(2n-8k,n-4*k).

A360309 a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2k,n-3k) * binomial(2*k,k).

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

A383581 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2k,k) binomial(2(n-3k),n-3*k).