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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A376826 Array read by antidiagonals: T(n,k) = n! * [x^n] exp(2*x + (k/2)*x^2), n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 5, 8, 1, 2, 6, 14, 16, 1, 2, 7, 20, 43, 32, 1, 2, 8, 26, 76, 142, 64, 1, 2, 9, 32, 115, 312, 499, 128, 1, 2, 10, 38, 160, 542, 1384, 1850, 256, 1, 2, 11, 44, 211, 832, 2809, 6512, 7193, 512, 1, 2, 12, 50, 268, 1182, 4864, 15374, 32400, 29186, 1024
Offset: 0

Views

Author

Andrew Howroyd, Oct 07 2024

Keywords

Examples

			Array begins:
======================================================
n\k |   0    1    2     3     4     5     6      7 ...
----+-------------------------------------------------
  0 |   1    1    1     1     1     1     1      1 ...
  1 |   2    2    2     2     2     2     2      2 ...
  2 |   4    5    6     7     8     9    10     11 ...
  3 |   8   14   20    26    32    38    44     50 ...
  4 |  16   43   76   115   160   211   268    331 ...
  5 |  32  142  312   542   832  1182  1592   2062 ...
  6 |  64  499 1384  2809  4864  7639 11224  15709 ...
  7 | 128 1850 6512 15374 29696 50738 79760 118022 ...
     ...

Links

Crossrefs

Columns 0..5 are A000079, A005425, A000898, A202830, A193778, A202832.
Cf. A359762, A373625.

Programs

  • PARI
    T(n,k) = {sum(i=0, n\2, binomial(n,2*i) * 2^(n-2*i) * k^i * (2*i)!/(2^i*i!))}

Formula

E.g.f. of column k: exp(2*x + k*x^2/2).
Column k is the binomial transform of column k of A359762.
T(n,k) = Sum_{i=0..floor(n/2)} binomial(n,2*i) * 2^(n-2*i) * k^i * (2*i-1)!!.
T(n,k) = Sum_{i=0..floor(n/2)} 2^(n-3*i) * k^i * n! / ((n-2*i)! * i!).

A193777 Number of signed permutations of size 2n invariant under D and D'bar and avoiding (-2, 1) and (2, -1).

Original entry on oeis.org

1, 2, 6, 18, 58, 190, 642, 2206, 7746, 27662, 100738, 373550, 1413506, 5457710, 21546466, 87025806, 360264258, 1529624366, 6669850466, 29877013902, 137560725890, 650780790894, 3162711095074, 15774862353614, 80687636530882, 422713072650286, 2265833731786594
Offset: 0

Views

Author

Andy Hardt, Aug 04 2011

Keywords

Comments

Also the number of signed permutations of size n invariant under D and avoiding (-2, 1) and (2, -1).

Links

Crossrefs

Cf. A193778, A000085.

Programs

  • Maple
    inv := proc(n) option remember; if n<2 then 1 else inv(n-1)+(n-1)*inv(n-2) fi end:
a := proc(n) option remember; if n < 2 then n+1 else
2*a(n-1) + n*a(n-2) - add(j*a(j)*inv(n-j-2), j=1..n-3) fi end:
seq(a(n), n=0..26); # Peter Luschny, Dec 09 2018
  • Mathematica
    inv[n_] := inv[n] = If[n<2, 1, inv[n-1] + (n-1) inv[n-2]];
a[n_] := a[n] = If[n<2, n+1, 2 a[n-1] + n a[n-2] - Sum[j a[j] inv[n-j-2], {j, 1, n-3}]];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jun 26 2019, after Peter Luschny *)

Formula

a(n) = 2*a(n-1) + n*a(n-2) - Sum_{j=1..k-3} j*a(j)*|S_{k-j-3}^D|, where S_n^D is the set of unsigned permutations of length n invariant under D.
a(n) = 2*a(n-1) + n*a(n-2) - Sum_{j=1..n-3} j*a(j)*A000085(n-j-2). - Andrew Howroyd, Dec 09 2018

Extensions

Terms a(9) and beyond from Peter Luschny, Dec 09 2018
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