cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, May 06 2020

Keywords

Examples

			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, ...

Links

Crossrefs

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

Formula

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

A334569 E.g.f.: exp(-(x + x^2/2 + x^3/3)).

Original entry on oeis.org

1, -1, 0, 0, 6, -6, -24, -120, 540, 1764, 2016, -68256, -147960, 700920, 11870496, 5245344, -330495984, -2602348560, 6794046720, 141179998464, 619736321376, -6025074044256, -66284988059520, -87481563442560, 4660723755205056, 34028147176271424, -98057302990861824
Offset: 0

Views

Author

Seiichi Manyama, May 06 2020

Keywords

Links

Crossrefs

Column 3 of A334568.
Cf. A057693, A334562.

Programs

  • Mathematica
    m = 25; Range[0, m]! * CoefficientList[Series[Exp[-(x + x^2/2 + x^3/3)], {x, 0, m}], x] (* Amiram Eldar, May 03 2021 *)
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(-x-x^2/2-x^3/3)))

Formula

a(0) = 1 and a(n) = - (n-1)! * Sum_{k=1..min(3,n)} a(n-k)/(n-k)!.
D-finite with recurrence a(n) +a(n-1) +(n-1)*a(n-2) +(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, May 07 2020

A334570 E.g.f.: exp(-(x + x^2/2 + x^3/3 + x^4/4)).

Original entry on oeis.org

1, -1, 0, 0, 0, 24, -24, -120, -720, -5040, 32256, 169344, 743040, -380160, -91048320, -408710016, -174230784, 29842525440, 455765667840, 1061427732480, -18914141343744, -379628241094656, -3305984467691520, 10588546824468480, 458522651844956160, 5722461010387021824
Offset: 0

Views

Author

Seiichi Manyama, May 06 2020

Keywords

Links

Crossrefs

Column 4 of A334568.
Cf. A070945, A334564.

Programs

  • Mathematica
    m = 25; Range[0, m]! * CoefficientList[Series[Exp[-(x + x^2/2 + x^3/3 + x^4/4)], {x, 0, m}], x] (* Amiram Eldar, May 03 2021 *)
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(-x-x^2/2-x^3/3-x^4/4)))

Formula

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

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

Views

Author

Seiichi Manyama, May 06 2020

Keywords

Links

Crossrefs

Column 5 of A334568.
Cf. A070946, A334565.

Programs

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

Formula

a(0) = 1 and a(n) = - (n-1)! * Sum_{k=1..min(5,n)} a(n-k)/(n-k)!.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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