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 1-6 of 6 results.

A362277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (-k/2)^j * binomial(n-j,j)/(n-j)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -2, 1, 1, 1, -2, -5, -2, 1, 1, 1, -3, -8, 1, 6, 1, 1, 1, -4, -11, 10, 41, 16, 1, 1, 1, -5, -14, 25, 106, 31, -20, 1, 1, 1, -6, -17, 46, 201, -44, -461, -132, 1, 1, 1, -7, -20, 73, 326, -299, -1952, -895, 28, 1, 1, 1, -8, -23, 106, 481, -824, -5123, -1028, 6481, 1216, 1
Offset: 0

Views

Author

Seiichi Manyama, Apr 13 2023

Keywords

Examples

			Square array begins:
  1,  1,  1,   1,    1,    1,     1, ...
  1,  1,  1,   1,    1,    1,     1, ...
  1,  0, -1,  -2,   -3,   -4,    -5, ...
  1, -2, -5,  -8,  -11,  -14,   -17, ...
  1, -2,  1,  10,   25,   46,    73, ...
  1,  6, 41, 106,  201,  326,   481, ...
  1, 16, 31, -44, -299, -824, -1709, ...

Links

Crossrefs

Columns k=0..6 give A000012, (-1)^n * A001464(n), A293604, A362278, A362176, A362279, A362177.
Main diagonal gives A362276.
T(n,2*n) gives A362282.
Cf. A359762, A362302.

Programs

  • PARI
    T(n,k) = n!*sum(j=0,n\2, (-k/2)^j/(j!*(n-2*j)!));

Formula

E.g.f. of column k: exp(x - k*x^2/2).
T(n,k) = T(n-1,k) - k*(n-1)*T(n-2,k) for n > 1.
T(n,k) = n! * Sum_{j=0..floor(n/2)} (-k/2)^j / (j! * (n-2*j)!).

A362043 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * binomial(n-2*j,j)/(n-2*j)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 4, 9, 11, 1, 1, 1, 1, 5, 13, 21, 31, 1, 1, 1, 1, 6, 17, 31, 81, 106, 1, 1, 1, 1, 7, 21, 41, 151, 351, 337, 1, 1, 1, 1, 8, 25, 51, 241, 736, 1233, 1205, 1, 1, 1, 1, 9, 29, 61, 351, 1261, 2689, 5769, 5021, 1
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2023

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,  2,  3,   4,   5,   6,   7, ...
  1,  5,  9,  13,  17,  21,  25, ...
  1, 11, 21,  31,  41,  51,  61, ...
  1, 31, 81, 151, 241, 351, 481, ...

Links

Crossrefs

Columns k=0..2 give A000012, A190865, A001470.
Main diagonal gives A362173.
T(n,2*n) gives A362300.
T(n,6*n) gives A362301.
Cf. A359762, A362302.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\3, (k/6)^j/(j!*(n-3*j)!));

Formula

E.g.f. of column k: exp(x + k*x^3/6).
T(n,k) = T(n-1,k) + k * binomial(n-1,2) * T(n-3,k) for n > 2.
T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j / (j! * (n-3*j)!).

A362303 a(n) = n! * Sum_{k=0..floor(n/3)} (-n/6)^k * binomial(n-2*k,k)/(n-2*k)!.

Original entry on oeis.org

1, 1, 1, -2, -15, -49, 241, 3186, 17473, -136835, -2591199, -19940194, 214217521, 5280969123, 52303886545, -714177220574, -21687847310079, -262685369226919, 4351534043729473, 157014580915662750, 2248361900084617201, -43790588385118719689
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2023

Keywords

Links

Crossrefs

Main diagonal of A362302.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((2*lambertw(x^3/2))^(1/3))/(1+lambertw(x^3/2))))

Formula

a(n) = n! * [x^n] exp(x - n*x^3/6).
E.g.f.: exp( ( 2*LambertW(x^3/2) )^(1/3) ) / (1 + LambertW(x^3/2)).

A362304 a(n) = n! * Sum_{k=0..floor(n/3)} (-n/3)^k * binomial(n-2*k,k)/(n-2*k)!.

Original entry on oeis.org

1, 1, 1, -5, -31, -99, 1201, 13231, 70785, -1362311, -21562399, -161746749, 4263108961, 87979472725, 849097038609, -28416142768649, -723086288422399, -8532476619366159, 346207723221680065, 10474480743776327179, 146105160034616914401
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2023

Keywords

Links

Crossrefs

Cf. A362300, A362302.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(x^3)^(1/3))/(1+lambertw(x^3))))

Formula

a(n) = A362302(n,2*n).
a(n) = n! * [x^n] exp(x - n*x^3/3).
E.g.f.: exp( ( LambertW(x^3) )^(1/3) ) / (1 + LambertW(x^3)).

A362305 a(n) = n! * Sum_{k=0..floor(n/3)} (-n)^k * binomial(n-2*k,k)/(n-2*k)!.

Original entry on oeis.org

1, 1, 1, -17, -95, -299, 12241, 122011, 642433, -41645015, -597247199, -4407324569, 390913189921, 7315513279933, 69439658097265, -7816418805235949, -180448412456686079, -2093964182367814319, 285679499679525805633, 7844019340520912230495
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2023

Keywords

Links

Crossrefs

Cf. A362301, A362302.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((lambertw(3*x^3)/3)^(1/3))/(1+lambertw(3*x^3))))

Formula

a(n) = A362302(n,6*n).
a(n) = n! * [x^n] exp(x - n*x^3).
E.g.f.: exp( ( LambertW(3*x^3)/3 )^(1/3) ) / (1 + LambertW(3*x^3)).

A362309 Expansion of e.g.f. exp(x - x^3/3).

Original entry on oeis.org

1, 1, 1, -1, -7, -19, 1, 211, 1009, 953, -14239, -105049, -209879, 1669669, 18057313, 56255291, -294375199, -4628130319, -19929569471, 70149241423, 1652969810521, 9226206209501, -20236475188159, -783908527648861, -5452368869656367
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2023

Keywords

Links

Crossrefs

Column k=2 of A362302.
Cf. A001470, A246607.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-x^3/3)))

Formula

a(n) = a(n-1) - 2 * binomial(n-1,2) * a(n-3) for n > 2.
a(n) = n! * Sum_{k=0..floor(n/3)} (-1/3)^k / (k! * (n-3*k)!).
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.