A334565 -id:A334565 - cp's OEIS frontend

A334561 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j).

1, 1, -1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, 5, 1, 1, -1, -1, -1, 1, -1, 1, -1, -1, -1, 25, -41, 1, 1, -1, -1, -1, 1, 19, 31, -1, 1, -1, -1, -1, 1, 139, -209, 461, 1, 1, -1, -1, -1, 1, 19, 151, -2269, -895, -1, 1, -1, -1, -1, 1, 19, 871, -1429, 2801, -6481, 1, 1, -1, -1, -1, 1, 19, 151, 1091, -19039, 68615, 22591, -1

Offset: 0

Seiichi Manyama, May 06 2020

			Square array begins:
   1,   1,    1,   1,   1,   1,   1, ...
  -1,  -1,   -1,  -1,  -1,  -1,  -1, ...
   1,  -1,   -1,  -1,  -1,  -1,  -1, ...
  -1,   5,   -1,  -1,  -1,  -1,  -1, ...
   1,   1,   25,   1,   1,   1,   1, ...
  -1, -41,   19, 139,  19,  19,  19, ...
   1,  31, -209, 151, 871, 151, 151, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=1..5 give A033999, A000321, A334562, A334564, A334565.
Main diagonal gives A293116.
Cf. A293669, A334568.

A(0,k) = 1 and A(n,k) = - (n-1)! * Sum_{j=1..min(k,n)} j*A(n-j,k)/(n-j)!.

A334571 E.g.f.: exp(-(x + x^2/2 + x^3/3 + x^4/4 + x^5/5)).

Original entry on oeis.org

1, -1, 0, 0, 0, 0, 120, -120, -720, -5040, -40320, -362880, 3024000, 21859200, 151113600, 753667200, -3978374400, -249882796800, -1752623308800, -7319841177600, 79100731699200, 2905096654540800, 48387220210944000, 226481995918080000, -2094278695939584000, -85503113536135680000

Offset: 0

Seiichi Manyama, May 06 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..500

Column 5 of A334568.

Cf. A070946, A334565.

m = 25; Range[0, m]! * CoefficientList[Series[Exp[-(x + x^2/2 + x^3/3 + x^4/4 + x^5/5)], {x, 0, m}], x] (* Amiram Eldar, May 03 2021 *)

N=40; x='x+O('x^N); Vec(serlaplace(exp(-x-x^2/2-x^3/3-x^4/4-x^5/5)))

a(0) = 1 and a(n) = - (n-1)! * Sum_{k=1..min(5,n)} a(n-k)/(n-k)!.

cp's OEIS Frontend

A334561 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula