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.

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A247505 Generalized Lucas numbers: square array A(n,k) read by antidiagonals, A(n,k)=(-1)^(k+1)*k*[x^k](-log((1+sum_{j=1..n}(-1)^(j+1)*x^j)^(-1))), (n>=0, k>=0).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 3, 1, 0, 0, 1, 3, 4, 1, 0, 0, 1, 3, 7, 7, 1, 0, 0, 1, 3, 7, 11, 11, 1, 0, 0, 1, 3, 7, 15, 21, 18, 1, 0, 0, 1, 3, 7, 15, 26, 39, 29, 1, 0, 0, 1, 3, 7, 15, 31, 51, 71, 47, 1, 0, 0, 1, 3, 7, 15, 31, 57, 99, 131, 76, 1, 0
Offset: 0

Views

Author

Peter Luschny, Nov 02 2014

Keywords

Examples

			n\k[0][1][2][3] [4] [5] [6]  [7]  [8]  [9]  [10]  [11]  [12]
[0] 0, 0, 0, 0,  0,  0,  0,   0,   0,   0,    0,    0,    0
[1] 0, 1, 1, 1,  1,  1,  1,   1,   1,   1,    1,    1,    1
[2] 0, 1, 3, 4,  7, 11, 18,  29,  47,  76,  123,  199,  322 [A000032]
[3] 0, 1, 3, 7, 11, 21, 39,  71, 131, 241,  443,  815, 1499 [A001644]
[4] 0, 1, 3, 7, 15, 26, 51,  99, 191, 367,  708, 1365, 2631 [A073817]
[5] 0, 1, 3, 7, 15, 31, 57, 113, 223, 439,  863, 1695, 3333 [A074048]
[6] 0, 1, 3, 7, 15, 31, 63, 120, 239, 475,  943, 1871, 3711 [A074584]
[7] 0, 1, 3, 7, 15, 31, 63, 127, 247, 493,  983, 1959, 3903 [A104621]
[8] 0, 1, 3, 7, 15, 31, 63, 127, 255, 502, 1003, 2003, 3999 [A105754]
[.] .  .  .  .   .   .   .    .    .    .     .     .     .
oo] 0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095 [A000225]
'
As a triangular array, starts:
0,
0, 0,
0, 1, 0,
0, 1, 1, 0,
0, 1, 3, 1, 0,
0, 1, 3, 4, 1, 0,
0, 1, 3, 7, 7, 1,  0,
0, 1, 3, 7, 11, 11, 1, 0,
0, 1, 3, 7, 15, 21, 18, 1, 0,
0, 1, 3, 7, 15, 26, 39, 29, 1, 0,

Crossrefs

Cf. A247506, A000225, A000032, A001644, A073817, A074048, A074584, A104621, A105754.
Cf. A125127.

Programs

  • Maple
    A := proc(n, k) f := -log((1+add((-1)^(j+1)*x^j, j=1..n))^(-1));
(-1)^(k+1)*k*coeff(series(f,x,k+2),x,k) end:
seq(print(seq(A(n,k), k=0..12)), n=0..8);
  • Mathematica
    A[n_, k_] := Module[{f, x}, f = -Log[(1+Sum[(-1)^(j+1) x^j, {j, 1, n}] )^(-1)]; (-1)^(k+1) k SeriesCoefficient[f, {x, 0, k}]];
Table[A[n-k, k], {n, 0, 12}, {k, 0, n}] (* Jean-François Alcover, Jun 28 2019, from Maple *)

A247504 Square array read by antidiagonals, A(n,k) = k!*[x^k]((1-Sum_{j=1..n} x^j)^(-1)), (n>=0,k>=0).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 4, 6, 0, 1, 1, 4, 18, 24, 0, 1, 1, 4, 24, 120, 120, 0, 1, 1, 4, 24, 168, 960, 720, 0, 1, 1, 4, 24, 192, 1560, 9360, 5040, 0, 1, 1, 4, 24, 192, 1800, 17280, 105840, 40320, 0, 1, 1, 4, 24, 192, 1920, 20880, 221760, 1370880, 362880, 0
Offset: 0

Views

Author

Peter Luschny, Nov 03 2014

Keywords

Examples

			n\k[0][1][2] [3]  [4]   [5]    [6]     [7]      [8]       [9]
[0] 1, 0, 0,  0,   0,    0,     0,      0,       0,        0
[1] 1, 1, 2,  6,  24,  120,   720,   5040,   40320,   362880 [A000142]
[2] 1, 1, 4, 18, 120,  960,  9360, 105840, 1370880, 19958400 [A005442]
[3] 1, 1, 4, 24, 168, 1560, 17280, 221760, 3265920, 54069120
[.] .  .  .  .  .   .   .   .    .    .    .     .     .
oo] 1, 1, 4, 24, 192, 1920, 23040, 322560, 5160960, 92897280 [A002866]
'
As a triangular array, starts:
1,
1, 0,
1, 1, 0,
1, 1, 2, 0,
1, 1, 4, 6,  0,
1, 1, 4, 18, 24,  0,
1, 1, 4, 24, 120, 120, 0,
1, 1, 4, 24, 168, 960, 720, 0.

Crossrefs

Cf. A247505, A247506, A002866, A000142, A005442.

Programs

  • Maple
    A := (n,k) -> k!*coeff(series((1-add(x^j, j=1..n))^(-1),x,k+2),x,k): seq(print(seq(A(n,k), k=0..9)), n=0..7);
Showing 1-2 of 2 results.

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

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