A360186 -id:A360186 - cp's OEIS frontend

A307354 a(n) = Sum_{0<=i<=j<=n} (-1)^(i+j) * (i+j)!/(i!*j!).

1, 2, 6, 19, 65, 231, 841, 3110, 11628, 43834, 166298, 634140, 2428336, 9331688, 35967462, 138987715, 538287881, 2088842463, 8119916647, 31613327405, 123251518641, 481125828853, 1880262896537, 7355767408395, 28803717914791, 112887697489907, 442784607413427

Offset: 0

Seiichi Manyama, Apr 03 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1664

Partial sums of A026641. - Seiichi Manyama, Jan 30 2023
Cf. A000108, A006134, A079309, A105872, A120305, A306409.
Cf. A191993, A360186, A360212.

Table[Sum[Sum[(-1)^(i + j)*(i + j)!/(i!*j!), {i, 0, j}], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Apr 04 2019 *)

a(n) = sum(i=0, n, sum(j=i, n, (-1)^(i+j)*(i+j)!/(i!*j!)));

a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-3*k, n)); \\ Seiichi Manyama, Jan 29 2023

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+x^3*(2/(1+sqrt(1-4*x)))^3))) \\ Seiichi Manyama, Jan 29 2023

a(n) = (A006134(n) + A120305(n))/2.
From Vaclav Kotesovec, Apr 04 2019: (Start)
Recurrence: 2*n*a(n) = (9*n-4)*a(n-1) - (3*n-2)*a(n-2) - 2*(2*n-1)*a(n-3).
a(n) ~ 2^(2*n+3) / (9*sqrt(Pi*n)). (End)
From Seiichi Manyama, Jan 29 2023: (Start)
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-3*k,n).
G.f.: 1 / ( sqrt(1-4*x) * (1 + x^3 * c(x)^3) ), where c(x) is the g.f. of A000108. (End)
a(n) = [x^n] 1/((1+x^3) * (1-x)^(n+1)). - Seiichi Manyama, Apr 08 2024

A360185 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2n-4k,n-2*k).

Original entry on oeis.org

1, 2, 5, 18, 65, 234, 859, 3198, 12011, 45422, 172745, 660010, 2531411, 9740590, 37585189, 145376930, 563495201, 2188229290, 8511640099, 33157034510, 129334888721, 505100839930, 1974764074999, 7728329887670, 30272839608101, 118682276550082, 465645693340003

Offset: 0

Seiichi Manyama, Jan 29 2023

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

Cf. A054108, A360186.

A360185 := proc(n)
    add((-1)^k*binomial(2*n-4*k,n-2*k),k=0..n/2) ;
end proc:
seq(A360185(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

a(n) = sum(k=0, n\2, (-1)^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) ).
a(n) ~ 2^(2*n + 4) / (17*sqrt(Pi*n)). - Vaclav Kotesovec, Jan 29 2023
D-finite with recurrence n*a(n) +2*(-2*n+1)*a(n-1) +n*a(n-2) +2*(-2*n+1)*a(n-3)=0. - R. J. Mathar, Mar 12 2023
a(n)+a(n-2) = A000984(n). - R. J. Mathar, Mar 12 2023

A360291 a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2k,k) binomial(2n-6k,n-3*k).

Original entry on oeis.org

1, 2, 6, 20, 72, 264, 984, 3714, 14148, 54284, 209482, 812196, 3161340, 12345658, 48348522, 189807336, 746740510, 2943359208, 11620961412, 45950375602, 181936110006, 721233025332, 2862271873966, 11370584735100, 45212101270728, 179926167512914

Offset: 0

Seiichi Manyama, Feb 01 2023

nonn

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

Cf. A085362, A360290, A360292.

Cf. A360186, A360294, A383581.

a(n) = sum(k=0, n\3, 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).
a(n) = A383581(n) - A383581(n-3). - Seiichi Manyama, May 01 2025

A360212 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2n-5k,n-3*k).

Original entry on oeis.org

1, 2, 6, 19, 67, 242, 890, 3310, 12423, 46959, 178526, 681893, 2614698, 10059000, 38807021, 150080294, 581649776, 2258469988, 8783966719, 34214789901, 133450049457, 521134066663, 2037313708685, 7972641631438, 31228124666374, 122421230120657

Offset: 0

Seiichi Manyama, Jan 30 2023

nonn

Cf. A191993, A307354, A360186.

Cf. A000108, A360152.

A360212 := proc(n)
    add((-1)^k*binomial(2*n-5*k,n-3*k),k=0..n/3) ;
end proc:
seq(A360212(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-5*k, n-3*k));

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

G.f.: 1 / ( sqrt(1-4*x) * (1 + x^3 * c(x)) ), where c(x) is the g.f. of A000108.

D-finite with recurrence 2*n*a(n) +4*(-2*n+1)*a(n-1) +(3*n-4)*a(n-2) +2*(-6*n+11)*a(n-3) +(n-4)*a(n-4) +2*(-n+9)*a(n-5) +4*(-2*n+1)*a(n-6) +(n-4)*a(n-7) +2*(-2*n+9)*a(n-8)=0. - R. J. Mathar, Mar 12 2023

cp's OEIS Frontend

A307354 a(n) = Sum_{0<=i<=j<=n} (-1)^(i+j) * (i+j)!/(i!*j!).

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A360185 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-4*k,n-2*k).

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A360291 a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2*k,k) * binomial(2*n-6*k,n-3*k).

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Formula

A360212 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-5*k,n-3*k).

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Author

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Programs

Maple

PARI

PARI

Formula

A360185 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2n-4k,n-2*k).

A360291 a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2k,k) binomial(2n-6k,n-3*k).

A360212 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2n-5k,n-3*k).