A307354 -id:A307354 - cp's OEIS frontend

A120305 a(n) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j) * (i+j)!/(i!j!).

1, 1, 3, 9, 31, 111, 407, 1513, 5679, 21471, 81643, 311895, 1196131, 4602235, 17757183, 68680169, 266200111, 1033703055, 4020716123, 15662273839, 61092127491, 238582873475, 932758045123, 3650336341239, 14298633670931

Offset: 0

Alexander Adamchuk, Jul 14 2006

nonn

p divides a((p+1)/2) for prime p = 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, ... (A033200: primes congruent to {1, 3} mod 8; or, odd primes of the form x^2 + 2*y^2).
p divides a((p-3)/2) for prime p = 17, 41, 73, 89, 97, 113, 137, ... (A007519: primes of the form 8n+1).
Essentially the same as partial sums of A072547. - Seiichi Manyama, Jan 30 2023

Seiichi Manyama, Table of n, a(n) for n = 0..1664 (terms 0..200 from Vincenzo Librandi)

Cf. A000108, A006134, A033200, A007519, A092785, A307354.

Cf. A072547, A371798.

Table[Sum[Sum[(-1)^(i+j)*(i+j)!/(i!j!),{i,0,n}],{j,0,n}],{n,0,50}]

a(n) = sum(i=0, n, sum(j=0, n, (-1)^(i+j) * (i+j)!/(i!*j!))); \\ Michel Marcus, Apr 02 2019

a(n) = sum(i=0, 2*n, (-1)^i*i!*polcoef(sum(j=0, n, x^j/j!)^2, i)); \\ Seiichi Manyama, May 20 2019

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

a(n) = Sum_{j=0..n} Sum_{i=0..n} (-1)^(i+j)*(i+j)!/(i!j!).
Recurrence: 2*n*(3*n-5)*a(n) = 3*(9*n^2 - 19*n + 8)*a(n-1) - 3*(n-1)*(3*n-4)*a(n-2) - 2*(2*n-3)*(3*n-2)*a(n-3). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 4^(n+1)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 13 2013
G.f.: ( 1/(sqrt(1-4*x) * (1-x)) ) * ( (1 - x *c(x))/(1 + x *c(x)) ), where c(x) is the g.f. of A000108. - Seiichi Manyama, Jan 30 2023
From Seiichi Manyama, Apr 06 2024: (Start)
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-3*k-1,n-3*k).
a(n) = [x^n] 1/((1+x^3) * (1-x)^n). (End)

A306409 a(n) = -Sum_{0<=i

Original entry on oeis.org

0, 1, 3, 10, 34, 120, 434, 1597, 5949, 22363, 84655, 322245, 1232205, 4729453, 18210279, 70307546, 272087770, 1055139408, 4099200524, 15951053566, 62159391150, 242542955378, 947504851414, 3705431067156, 14505084243860, 56831711106496, 222853334131080

Offset: 0

Seiichi Manyama, Apr 05 2019

nonn

			n | a(n) | A307354 | A006134 | A120305
--+------+---------+---------+---------
0 |    0 |       1 |       1 |       1
1 |    1 |       2 |       3 |       1
2 |    3 |       6 |       9 |       3
3 |   10 |      19 |      29 |       9
4 |   34 |      65 |      99 |      31
5 |  120 |     231 |     351 |     111

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

Partial sums of A014300. - Seiichi Manyama, Jan 30 2023

Cf. A000108, A000957, A006134, A057552, A120305, A307354.

Table[-Sum[Sum[(-1)^(i+j) * (i+j)!/(i!*j!), {i, 0, j-1}], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Apr 05 2019 *)

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

my(N=30, x='x+O('x^N)); concat(0, Vec((1-sqrt(1-4*x))/(sqrt(1-4*x)*(1-x)*(3-sqrt(1-4*x))))) \\ Seiichi Manyama, Jan 30 2023

a(n) = A006134(n) - A307354(n).
a(n) = (A006134(n) - A120305(n))/2.
a(n) ~ 4^(n+1) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 05 2019
G.f.: ( 1/(sqrt(1-4*x) * (1-x)) ) * ( x *c(x)/(1 + x *c(x)) ), where c(x) is the g.f. of A000108. - Seiichi Manyama, Jan 30 2023

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

A307358 a(n) = Sum_{0<=i<=j<=k<=n} (-1)^(i+j+k) * (i+j+k)!/(i!j!k!).

Original entry on oeis.org

1, -4, 72, -1345, 27886, -610558, 13861334, -322838475, 7663363513, -184598740512, 4498935186891, -110693299767349, 2745124008220296, -68532209858173364, 1720678086867077832, -43415209670536390089, 1100146390869600888470

Offset: 0

Seiichi Manyama, Apr 04 2019

sign

Cf. A144660, A307318, A307352, A307354.

Table[Sum[Sum[Sum[(-1)^(i+j+k) * (i+j+k)!/(i!*j!*k!), {i, 0, j}], {j, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 04 2019 *)

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

a(n) ~ (-1)^n * 3^(3*n + 13/2) / (1792*Pi*n). - Vaclav Kotesovec, Apr 04 2019

cp's OEIS Frontend

A120305 a(n) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j) * (i+j)!/(i!j!).

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A306409 a(n) = -Sum_{0<=i

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

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A307358 a(n) = Sum_{0<=i<=j<=k<=n} (-1)^(i+j+k) * (i+j+k)!/(i!*j!*k!).

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Programs

Mathematica

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Formula

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

A307358 a(n) = Sum_{0<=i<=j<=k<=n} (-1)^(i+j+k) * (i+j+k)!/(i!j!k!).