A371777 -id:A371777 - cp's OEIS frontend

A371773 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-k+1,n-3k).

1, 3, 10, 36, 134, 507, 1937, 7449, 28783, 111623, 434130, 1692387, 6610292, 25861384, 101319095, 397428091, 1560588454, 6133768656, 24128550045, 94986663925, 374188128311, 1474980414870, 5817387549611, 22955930045826, 90629404431826, 357960414264163

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

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

Cf. A371774, A371775, A371776.

Cf. A144904, A371758, A371777.

Table[Sum[Binomial[2n-k+1,n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* Harvey P. Dale, Sep 09 2024 *)

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

a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^n).
a(n) = binomial(2*(n-1), n-1)*hypergeom([1, (1-n)/3, (2-n)/3, 1-n/3], [1-n, 3/2-n, n], -27/4). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: n*(n^2 - 7)*a(n) = (9*n^3 - 2*n^2 - 79*n + 60)*a(n-1) - 2*(12*n^3 - 5*n^2 - 124*n + 150)*a(n-2) + (17*n^3 - 8*n^2 - 183*n + 240)*a(n-3) - 2*(2*n - 5)*(n^2 + 2*n - 6)*a(n-4).
a(n) ~ 2^(2*n+2) / sqrt(Pi*n). (End)

A371758 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-3k-1,n-3*k).

Original entry on oeis.org

1, 1, 3, 11, 39, 141, 519, 1933, 7263, 27479, 104543, 399543, 1532779, 5899167, 22766607, 88073091, 341425551, 1326019653, 5158412943, 20096457549, 78396460299, 306190920837, 1197181197567, 4685523856881, 18354865147011, 71962695111841, 282357198103815

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A371770, A371771, A371772.

Cf. A144904, A371773, A371777.

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

a(n) = [x^n] 1/((1-x^3) * (1-x)^n).
a(n) = binomial(2*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-2*n)/3, 2*(1-n)/3, 1-2*n/3], 1). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 3*n*(7*n-11)*a(n) = 6*(2*n-3)*(7*n-4)*a(n-1) - n*(7*n-11)*a(n-2) + 2*(2*n-3)*(7*n-4)*a(n-3).
a(n) ~ 2^(2*n+2) / (7*sqrt(Pi*n)). (End)

A360168 a(n) = Sum_{k=0..floor(n/3)} binomial(2n,n-3k).

Original entry on oeis.org

1, 2, 6, 21, 78, 297, 1145, 4447, 17358, 68001, 267141, 1051767, 4148281, 16385111, 64797543, 256515731, 1016368078, 4030114641, 15990813773, 63485616391, 252175202373, 1002136689071, 3984080489263, 15844839393411, 63036297959993, 250855287692647

Offset: 0

Seiichi Manyama, Jan 28 2023

nonn

Cf. A105872, A144904, A360150, A360151, A360152, A360153.
Cf. A000108, A032443, A114121.
Cf. A002450, A371777.

A360168 := proc(n)
    add(binomial(2*n,n-3*k),k=0..n/3) ;
end proc:
seq(A360168(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

a[n_] := Sum[Binomial[2*n, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)

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

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3*(2/(1+sqrt(1-4*x)))^6)))

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)^6) ), where c(x) is the g.f. of A000108.
D-finite with recurrence n*a(n) +2*(-4*n+3)*a(n-1) +8*(2*n-3)*a(n-2) +3*(-n+2)=0. - R. J. Mathar, Mar 12 2023
a(n) = [x^n] 1/(((1-x)^3-x^3) * (1-x)^(n-2)). - Seiichi Manyama, Apr 10 2024

A371780 a(n) = Sum_{k=0..floor(n/3)} binomial(5n+2,n-3k).

Original entry on oeis.org

1, 7, 66, 681, 7337, 81081, 911153, 10361554, 118881714, 1373402934, 15954079557, 186165866937, 2180501226751, 25620628577083, 301858589475117, 3564841627421691, 42186363329210473, 500142626996777355, 5939062937833796486, 70626949319708756435

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Robert Israel, Table of n, a(n) for n = 0..918

Cf. A371777, A371778, A371779.

Cf. A371739.

f:= proc(n) local k; add(binomial(5*n+2,n-3*k),k=0..n/3); end proc:
map(f, [$0..100]); # Robert Israel, Apr 22 2024

a(n) = sum(k=0, n\3, binomial(5*n+2, n-3*k));

a(n) = [x^n] 1/(((1-x)^3-x^3) * (1-x)^(4*n)).

A371778 a(n) = Sum_{k=0..floor(n/3)} binomial(3n+2,n-3k).

Original entry on oeis.org

1, 5, 28, 166, 1015, 6324, 39901, 254035, 1628380, 10493680, 67914088, 441086947, 2873255906, 18763759019, 122803467241, 805241108334, 5288922607095, 34789875710568, 229147231044397, 1511104857207706, 9975701630282920, 65920216186587257

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A371777, A371779, A371780.

Cf. A066380.

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

a(n) = [x^n] 1/(((1-x)^3-x^3) * (1-x)^(2*n)).

A371779 a(n) = Sum_{k=0..floor(n/3)} binomial(4n+2,n-3k).

Original entry on oeis.org

1, 6, 45, 365, 3078, 26565, 232831, 2063235, 18435021, 165780758, 1498533273, 13603087800, 123920995101, 1132284232215, 10372554403620, 95233251146671, 876081280823430, 8073359613286509, 74513645742072841, 688682977876117698, 6373025238727622277

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A371777, A371778, A371780.

Cf. A066381.

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

a(n) = [x^n] 1/(((1-x)^3-x^3) * (1-x)^(3*n)).

A371820 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2n+2,n-3k).

Original entry on oeis.org

1, 4, 15, 55, 200, 726, 2640, 9636, 35343, 130339, 483395, 1802901, 6760781, 25482643, 96506229, 367077447, 1401772536, 5372120718, 20653929804, 79634421312, 307826528346, 1192608522258, 4629875048634, 18006340509702, 70142823370656, 273633773330844

Offset: 0

Seiichi Manyama, Apr 06 2024

nonn

Cf. A001791, A120305, A371818, A371819.

Cf. A371777.

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

a(n) = [x^n] 1/(((1-x)^3+x^3) * (1-x)^n).
a(n) = binomial(2*(1+n), n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [1+n/3, (4+n)/3, (5+n)/3], 1). - Stefano Spezia, Apr 07 2024
a(n) ~ 2^(2*n+1) / sqrt(Pi*n). - Vaclav Kotesovec, Apr 19 2024

A371873 a(n) = Sum_{k=0..floor(n/3)} binomial(2n+1,n-3k).

Original entry on oeis.org

1, 3, 10, 36, 135, 517, 2003, 7815, 30634, 120480, 475002, 1876294, 7422676, 29400192, 116567356, 462561572, 1836843591, 7298613997, 29016050831, 115408159467, 459209330821, 1827849895817, 7277945888781, 28986847296997, 115479393316211, 460159673245743

Offset: 0

Seiichi Manyama, Apr 10 2024

nonn

Cf. A360150, A371871, A371872.

Cf. A002450, A360168, A371777.

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

a(n) = [x^n] 1/(((1-x)^3-x^3) * (1-x)^(n-1)).

cp's OEIS Frontend

A371773 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-k+1,n-3*k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A371758 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-3*k-1,n-3*k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A360168 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n,n-3*k).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A371780 a(n) = Sum_{k=0..floor(n/3)} binomial(5*n+2,n-3*k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

A371778 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n+2,n-3*k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A371779 a(n) = Sum_{k=0..floor(n/3)} binomial(4*n+2,n-3*k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A371820 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n+2,n-3*k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A371873 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n+1,n-3*k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A371773 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-k+1,n-3k).

A371758 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-3k-1,n-3*k).

A360168 a(n) = Sum_{k=0..floor(n/3)} binomial(2n,n-3k).

A371780 a(n) = Sum_{k=0..floor(n/3)} binomial(5n+2,n-3k).

A371778 a(n) = Sum_{k=0..floor(n/3)} binomial(3n+2,n-3k).

A371779 a(n) = Sum_{k=0..floor(n/3)} binomial(4n+2,n-3k).

A371820 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2n+2,n-3k).

A371873 a(n) = Sum_{k=0..floor(n/3)} binomial(2n+1,n-3k).