A331689 -id:A331689 - cp's OEIS frontend

A331688 E.g.f.: exp(-x/(1 - x)) / (1 - 2*x).

1, 1, 3, 17, 137, 1389, 16819, 236557, 3792753, 68326073, 1366917731, 30074632521, 721798881913, 18766625660197, 525460685327187, 15763716503597189, 504436925448024929, 17150818356045629937, 617428780939911647683, 23462281235407345160833

Offset: 0

Ilya Gutkovskiy, Jan 24 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..403

Cf. A000165, A000166, A000354, A009940, A293116, A331689.

f:= gfun:-rectoproc({a(n) = -(n - 1)*(5*n - 8)*a(n - 2) + (-3 + 4*n)*a(n - 1) + 2*(n - 1)*(n - 2)^2*a(n - 3),a(0)=1,a(1)=1,a(2)=3},a(n),remember):
map(f, [$0..30]); # Robert Israel, Jul 28 2020

nmax = 19; CoefficientList[Series[Exp[-x/(1 - x)]/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 k! Subfactorial[n - k], {k, 0, n}], {n, 0, 19}]

a(n) = Sum_{k=0..n} binomial(n,k)^2 * k! * A000166(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * k! * 2^k * A293116(n-k).
a(n) ~ n! * exp(-1) * 2^n. - Vaclav Kotesovec, Jan 26 2020
a(n) = (4*n-3)*a(n-1)-(n-1)*(5*n-8)*a(n-2)+2*(n-1)*(n--2)^2*a(n-3). - Robert Israel, Jul 28 2020

A367962 Triangle read by rows. T(n, k) = Sum_{j=0..k} (n!/j!).

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 6, 12, 15, 16, 24, 48, 60, 64, 65, 120, 240, 300, 320, 325, 326, 720, 1440, 1800, 1920, 1950, 1956, 1957, 5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700, 40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601

Offset: 0

Peter Luschny, Dec 06 2023

			  [0]   1;
  [1]   1,    2;
  [2]   2,    4,    5;
  [3]   6,   12,   15,   16;
  [4]  24,   48,   60,   64,   65;
  [5] 120,  240,  300,  320,  325,  326;
  [6] 720, 1440, 1800, 1920, 1950, 1956, 1957;

Cf. A094587, A000142 (T(n, 0)), A052849 (T(n, 1)), A000522 (T(n, n)), A007526 (T(n,n-1)), A038154 (T(n, n-2)), A355268 (T(n, n/2)), A367963(n) (T(2*n, n)/n!).

Cf. A001339 (row sums), A087208 (alternating row sums), A082030 (accumulated sums), A053482, A331689.

T := (n, k) -> add(n!/j!, j = 0..k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Module[{n=1},NestList[Append[n#,1+Last[#]n++]&,{1},10]] (* or *)
Table[Sum[n!/j!,{j,0,k}],{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 07 2023 *)

from functools import cache
@cache
def a_row(n: int) -> list[int]:
    if n == 0: return [1]
    row = a_row(n - 1) + [0]
    for k in range(n): row[k] *= n
    row[n] = row[n - 1] + 1
    return row

def T(n, k): return sum(falling_factorial(n, n - j) for j in range(k + 1))
for n in range(9): print([T(n, k) for k in range(n + 1)])

T(n, k) = A094587(n, k) * A000522(k).
T(n, k) = e * (n! / k!) * Gamma(k + 1, 1).
Sum_{k=0..n} T(n, k) * 2^(n - k) = A053482(n).
Sum_{k=0..n} T(n, k) * binomial(n, k) = A331689(n).
Recurrence: T(n, n) = T(n, n-1) + 1 starting with T(0, 0) = 1.
For k <> n: T(n, k) = n * T(n-1, k).

cp's OEIS Frontend

A331688 E.g.f.: exp(-x/(1 - x)) / (1 - 2*x).

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