A104557 Triangle T, read by rows, such that the unsigned columns of the matrix inverse when read downwards equals the rows of T read backwards, with T(n,n)=1 and T(n,n-1) = floor((n+1)/2)*floor((n+2)/2).

1, 1, 1, 2, 2, 1, 6, 6, 4, 1, 24, 24, 18, 6, 1, 120, 120, 96, 36, 9, 1, 720, 720, 600, 240, 72, 12, 1, 5040, 5040, 4320, 1800, 600, 120, 16, 1, 40320, 40320, 35280, 15120, 5400, 1200, 200, 20, 1, 362880, 362880, 322560, 141120, 52920, 12600, 2400, 300, 25, 1

Offset

0,4

Comments

Matrix inverse is A104558. Row sums form A102038. See A104559 for further formulas, where A104559(n,k) = T(n,k)/(n-k)!.

Links

Table of n, a(n) for n=0..54.

Formula

Formula: T(n,k) = (n-k)!*C(n-floor(k/2), floor((k+1)/2))*C(n-floor((k+1)/2), floor(k/2)).

Recurrence: T(n,k) = n*T(n-1,k) + T(n-2,k-2) for n >= k >= 2, with T(0,0) = T(1,0) = T(1,1) = 1.

T(n,0) = n!.

T(n,k) = T(n-1,k-1) + floor((k+2)/2)*T(n,k+1), T(0,0)=1, T(n,k)=0 for k > n or for k < 0. - Philippe Deléham, Dec 18 2006

Example

Rows of T begin:
      1;
      1,     1;
      2,     2,     1;
      6,     6,     4,     1;
     24,    24,    18,     6,    1;
    120,   120,    96,    36,    9,    1;
    720,   720,   600,   240,   72,   12,   1;
   5040,  5040,  4320,  1800,  600,  120,  16,  1;
  40320, 40320, 35280, 15120, 5400, 1200, 200, 20, 1; ...
The matrix inverse A104558 begins:
   1;
  -1,  1;
   0, -2,  1;
   0,  2, -4,   1;
   0,  0,  6,  -6,   1;
   0,  0, -6,  18,  -9,   1;
   0,  0,  0, -24,  36, -12,   1;
   0,  0,  0,  24, -96,  72, -16, 1; ...

Prog

(PARI) T(n, k)=(n-k)!*binomial(n-(k\2), (k+1)\2)*binomial(n-((k+1)\2), k\2)

Crossrefs

Cf. A104558, A104559, A102038.

Sequence in context: A121284 A225112 A108076 * A141712 A098539 A222073

Adjacent sequences: A104554 A104555 A104556 * A104558 A104559 A104560

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 16 2005

Status

approved