A368765 -id:A368765 - cp's OEIS frontend

A368766 a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+1,2) / k!).

1, 0, 3, 3, 22, 95, 591, 4109, 32908, 296127, 2961325, 32574509, 390894186, 5081624327, 71142740683, 1067141110125, 17074257762136, 290262381956159, 5224722875211033, 99269734629009437, 1985394692580188950, 41693288544183967719, 917252347972047290071

Offset: 0

Seiichi Manyama, Jan 04 2024

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..449

Cf. A368765, A368767, A368768.

Cf. A009574, A368762.

nxt[{n_,a_}]:={n+1,a(n+1)+(-1)^(n+1) Binomial[n+2,2]}; NestList[nxt,{0,1},30][[;;,2]] (* Harvey P. Dale, Mar 26 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x*sum(k=0, 1, binomial(1, k)*(-x)^k/(k+1)!)*exp(-x))/(1-x)))

a(0) = 1; a(n) = n*a(n-1) + (-1)^n * binomial(n+1,2).
a(n) = n! + (-1)^n * A009574(n).
E.g.f.: (1 - x * (1-x/2) * exp(-x)) / (1-x).

A368767 a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+2,3) / k!).

Original entry on oeis.org

1, 0, 4, 2, 28, 105, 686, 4718, 37864, 340611, 3406330, 37469344, 449632492, 5845221941, 81833107734, 1227496615330, 19639945846096, 333879079382663, 6009823428889074, 114186645148891076, 2283732902977823060, 47958390962534282489, 1055084601175754216782

Offset: 0

Seiichi Manyama, Jan 04 2024

nonn

Cf. A368765, A368766, A368768.

Cf. A368585, A368763.

my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x*sum(k=0, 2, binomial(2, k)*(-x)^k/(k+1)!)*exp(-x))/(1-x)))

a(0) = 1; a(n) = n*a(n-1) + (-1)^n * binomial(n+2,3).
a(n) = n! + (-1)^n * A368585(n).
E.g.f.: (1 - x * (1-x+x^2/6) * exp(-x)) / (1-x).

A368768 a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+3,4) / k!).

Original entry on oeis.org

1, 0, 5, 0, 35, 105, 756, 5082, 40986, 368379, 3684505, 40528554, 486344013, 6322470349, 88514587266, 1327718805930, 21243500898756, 361139515274007, 6500511274938111, 123509714223816794, 2470194284476344735, 51874079974003228809, 1141229759428071046448

Offset: 0

Seiichi Manyama, Jan 04 2024

nonn

Cf. A368765, A368766, A368767.

Cf. A368586, A368764.

my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x*sum(k=0, 3, binomial(3, k)*(-x)^k/(k+1)!)*exp(-x))/(1-x)))

a(0) = 1; a(n) = n*a(n-1) + (-1)^n * binomial(n+3,4).
a(n) = n! + (-1)^n * A368586(n).
E.g.f.: (1 - x * (1-3*x/2+x^2/2-x^3/24) * exp(-x)) / (1-x).

cp's OEIS Frontend

A368766 a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+1,2) / k!).

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A368767 a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+2,3) / k!).

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A368768 a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+3,4) / k!).

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