A334569 -id:A334569 - cp's OEIS frontend

A334568 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/j).

1, 1, -1, 1, -1, 1, 1, -1, 0, -1, 1, -1, 0, 2, 1, 1, -1, 0, 0, -2, -1, 1, -1, 0, 0, 6, -6, 1, 1, -1, 0, 0, 0, -6, 16, -1, 1, -1, 0, 0, 0, 24, -24, 20, 1, 1, -1, 0, 0, 0, 0, -24, -120, -132, -1, 1, -1, 0, 0, 0, 0, 120, -120, 540, -28, 1, 1, -1, 0, 0, 0, 0, 0, -120, -720, 1764, 1216, -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,    0,    0,    0,    0,    0,    0, ...
  -1,    2,    0,    0,    0,    0,    0, ...
   1,   -2,    6,    0,    0,    0,    0, ...
  -1,   -6,   -6,   24,    0,    0,    0, ...
   1,   16,  -24,  -24,  120,    0,    0, ...
  -1,   20, -120, -120, -120,  720,    0, ...
   1, -132,  540, -720, -720, -720, 5040, ...

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

Columns k=1..5 give A033999, A001464, A334569, A334570, A334571.

Cf. A330858, A334561.

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

A334562 E.g.f.: exp(-(x + x^2 + x^3)).

Original entry on oeis.org

1, -1, -1, -1, 25, 19, -209, -2269, 2801, 68615, 371071, -2499641, -28306871, -58645861, 1964456495, 15133179179, -37119981599, -1861550428529, -9225044407169, 110317002942095, 2150185424201081, 3953685082287779, -233260896605772881, -2920858244957587661, 7649165533910291665

Offset: 0

Seiichi Manyama, May 06 2020

sign

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

Column 3 of A334561.

Cf. A118589, A334569.

With[{nn=30},CoefficientList[Series[Exp[-(x+x^2+x^3)],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Nov 26 2020 *)

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

a(0) = 1 and a(n) = - (n-1)! * Sum_{k=1..min(3,n)} k*a(n-k)/(n-k)!.

D-finite with recurrence a(n) + a(n-1) +2*(n-1)*a(n-2) +3*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, May 07 2020

cp's OEIS Frontend

A334568 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/j).

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