A383379 -id:A383379 - cp's OEIS frontend

A295182 a(n) = n! * [x^n] exp(-n*x)/(1 - x)^n.

1, 0, 2, 6, 72, 620, 8640, 122346, 2156672, 41367672, 905126400, 21646532270, 570077595648, 16268377195044, 502096929431552, 16629319748711250, 588938142209310720, 22196966267762213744, 887352465220427317248, 37496112562144553167062, 1670071417348195942400000, 78195398849926292810318940

Offset: 0

Ilya Gutkovskiy, Nov 16 2017

nonn

The n-th term of the n-fold exponential convolution of A000166 with themselves.

Robert Israel, Table of n, a(n) for n = 0..396
N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Main diagonal of A295181.

Cf. A000166, A000407, A001813, A087981, A137775, A295183, A383379.

S:= series((exp(-x)/(1-x))^n,x,30):
seq(n!*coeff(S,x,n),n=0..29); # Robert Israel, Nov 16 2017

Table[n! SeriesCoefficient[Exp[-n x]/(1 - x)^n, {x, 0, n}], {n, 0, 21}]

a(n) = A295181(n,n).
a(n) ~ phi^(3*n - 1/2) * n^n / (5^(1/4) * exp(n*(1 + 1/phi))), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 16 2017
a(n) = n! * Sum_{k=0..n} (-n)^(n-k) * binomial(n+k-1,k)/(n-k)!. - Seiichi Manyama, Apr 25 2025

A383341 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * binomial(j+k,j)/(n-j)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 11, 24, 1, 1, 5, 16, 53, 120, 1, 1, 6, 21, 88, 309, 720, 1, 1, 7, 26, 129, 568, 2119, 5040, 1, 1, 8, 31, 176, 897, 4288, 16687, 40320, 1, 1, 9, 36, 229, 1296, 7317, 36832, 148329, 362880, 1, 1, 10, 41, 288, 1765, 11296, 67365, 354688, 1468457, 3628800

Offset: 0

Seiichi Manyama, Apr 24 2025

			Square array begins:
    1,    1,    1,    1,     1,     1,     1, ...
    1,    1,    1,    1,     1,     1,     1, ...
    2,    3,    4,    5,     6,     7,     8, ...
    6,   11,   16,   21,    26,    31,    36, ...
   24,   53,   88,  129,   176,   229,   288, ...
  120,  309,  568,  897,  1296,  1765,  2304, ...
  720, 2119, 4288, 7317, 11296, 16315, 22464, ...

Columns k=0..4 give A000142, A000255, A052124, A383378, A383383.
Main diagonal gives A383379.
Cf. A295181.

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

E.g.f. of column k: exp(-k*x) / (1-x)^(k+1).

A(0,k) = A(1,k) = 1; A(n,k) = n*A(n-1,k) + k*(n-1)*A(n-2,k).

cp's OEIS Frontend

A295182 a(n) = n! * [x^n] exp(-n*x)/(1 - x)^n.

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