A360144 -id:A360144 - cp's OEIS frontend

A360143 a(n) = Sum_{k=0..n} binomial(2n+2k,n-k).

1, 3, 13, 59, 271, 1250, 5775, 26696, 123423, 570576, 2637306, 12187755, 56312089, 260134905, 1201493926, 5548533913, 25619837773, 118283258215, 546041467522, 2520515546083, 11633752319476, 53693477980816, 247798435809211, 1143547904185879, 5277058908767419

Offset: 0

Seiichi Manyama, Jan 27 2023

nonn

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

Cf. A001700, A032443, A108080, A360144.

Cf. A000108, A000984, A002057.

A360143 := proc(n)
    add(binomial(2*n+2*k,n-k),k=0..n) ;
end proc:
seq(A360143(n),n=0..70) ;# R. J. Mathar, Mar 12 2023

Table[Sum[Binomial[2n+2k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jul 23 2025 *)

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

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

G.f.: 1 / ( sqrt(1-4*x) * (1 - x * c(x)^4) ), where c(x) is the g.f. of A000108.
D-finite with recurrence +n*(n-7)*a(n) -(7*n-4)*(n-7)*a(n-1) +4*(n^2-13*n+17)*a(n-2) +(35*n^2-217*n+304)*a(n-3) -2*(n-2)*(7*n-29)*a(n-4) +4*(n-2)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Mar 12 2023
a(n) = binomial(2*n, n)*hypergeom([1, -n, 1/2+n, 1+n], [(1+n)/3, (2+n)/3, 1+n/3], -4/27). - Stefano Spezia, Jun 17 2025

A360149 a(n) = Sum_{k=0..floor(n/2)} binomial(2n+k,n-2k).

Original entry on oeis.org

1, 2, 7, 27, 107, 429, 1731, 7012, 28478, 115864, 471991, 1924483, 7852083, 32053208, 130893949, 534673600, 2184482707, 8926392419, 36479840422, 149095843951, 609400587426, 2490900041118, 10181669553847, 41618414303969, 170118507902985, 695366323719302

Offset: 0

Seiichi Manyama, Jan 28 2023

nonn

Cf. A014300, A106188, A108081, A114121, A176287.

Cf. A000108, A360144, A360150.

A360149 := proc(n)
    add(binomial(2*n+k,n-2*k),k=0..floor(n/2)) ;
end proc:
seq(A360149(n),n=0..40) ; # R. J. Mathar, Mar 02 2023

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

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

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^2*(2/(1+sqrt(1-4*x)))^5)))

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^2 * c(x)^5) ), where c(x) is the g.f. of A000108.
a(n) ~ sqrt((7 - 5*(2/(173 + 21*sqrt(69)))^(1/3) + ((173 + 21*sqrt(69))/2)^(1/3)) / 69) / ((4 - (2/(25 - 3*sqrt(69)))^(1/3) - ((25 - 3*sqrt(69))/2)^(1/3))/3)^n. - Vaclav Kotesovec, Jan 28 2023
D-finite with recurrence n*(47*n-1011)*a(n) +(-261*n^2 +8567*n -6378)*a(n-1) +2*(-165*n^2 -9388*n +16143)*a(n-2) +(3089*n^2 +919*n -27492)*a(n-3) +2*(-1283*n^2 +3900*n +3981)*a(n-4) +4*(81*n+11)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Mar 02 2023

cp's OEIS Frontend

A360143 a(n) = Sum_{k=0..n} binomial(2n+2k,n-k).

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Formula

A360149 a(n) = Sum_{k=0..floor(n/2)} binomial(2n+k,n-2k).

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Maple

Mathematica

PARI

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cp's OEIS Frontend

A360143 a(n) = Sum_{k=0..n} binomial(2*n+2*k,n-k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A360149 a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k,n-2*k).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A360143 a(n) = Sum_{k=0..n} binomial(2n+2k,n-k).

A360149 a(n) = Sum_{k=0..floor(n/2)} binomial(2n+k,n-2k).