A368762 -id:A368762 - 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).

A368763 a(n) = n! * (1 + Sum_{k=0..n} binomial(k+2,3) / k!).

Original entry on oeis.org

1, 2, 8, 34, 156, 815, 4946, 34706, 277768, 2500077, 25000990, 275011176, 3300134476, 42901748643, 600624481562, 9009367224110, 144149875586576, 2450547884972761, 44109861929510838, 838087376660707252, 16761747533214146580, 351996698197497079951

Offset: 0

Seiichi Manyama, Jan 04 2024

nonn

Cf. A033540, A368762, A368764.

Cf. A368574, A368767.

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) + binomial(n+2,3).
a(n) = n! + A368574(n).
E.g.f.: (1 + x * (1+x+x^2/6) * exp(x)) / (1-x).

A368764 a(n) = n! * (1 + Sum_{k=0..n} binomial(k+3,4) / k!).

Original entry on oeis.org

1, 2, 9, 42, 203, 1085, 6636, 46662, 373626, 3363129, 33632005, 369953056, 4439438037, 57712696301, 807977750594, 12119666261970, 193914660195396, 3296549223326577, 59337886019884371, 1127419834377810364, 22548396687556216135, 473516330438680549461

Offset: 0

Seiichi Manyama, Jan 04 2024

nonn

Cf. A033540, A368762, A368763.

Cf. A368575, A368768.

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) + binomial(n+3,4).
a(n) = n! + A368575(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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A368763 a(n) = n! * (1 + Sum_{k=0..n} binomial(k+2,3) / k!).

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A368764 a(n) = n! * (1 + Sum_{k=0..n} binomial(k+3,4) / k!).

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