A334561 -id:A334561 - 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

A334564 E.g.f.: exp(-(x + x^2 + x^3 + x^4)).

Original entry on oeis.org

1, -1, -1, -1, 1, 139, 151, -1429, -19039, -170281, 517231, 12144439, 108063649, 294730019, -12338255929, -161010756061, -834576525119, 10909293532591, 350211422028511, 3176429907556079, -8268487440910399, -816508001025525061, -14017265801126968841, -55997107943920795301

Offset: 0

Seiichi Manyama, May 06 2020

sign

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

Column 4 of A334561.

Cf. A193930, A334570.

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

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

a(0) = 1 and a(n) = - (n-1)! * Sum_{k=1..min(4,n)} k*a(n-k)/(n-k)!.

A334565 E.g.f.: exp(-(x + x^2 + x^3 + x^4 + x^5)).

Original entry on oeis.org

1, -1, -1, -1, 1, 19, 871, 1091, -12319, -185401, -1871729, -16185401, 124125409, 2716194779, 30386265911, 192015557579, -1519087028159, -90118169425649, -1286566655151329, -8992926875821681, 96361313388009281, 5078872062174772259, 107995117552995689479, 996704518577739393619

Offset: 0

Seiichi Manyama, May 06 2020

sign

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

Column 5 of A334561.

Cf. A193931, A334571.

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

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

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

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A334564 E.g.f.: exp(-(x + x^2 + x^3 + x^4)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A334565 E.g.f.: exp(-(x + x^2 + x^3 + x^4 + x^5)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula