A373050 Triangle read by rows: T(n, k) = (Sum_{i=0..n-k} (-1)^i * binomial(n-k, i) * (n+2-i)!) * binomial(n, k) / ((k+1) * (k+2)) for 0 <= k <= n.

1, 2, 1, 7, 6, 2, 32, 39, 24, 6, 181, 284, 252, 120, 24, 1214, 2325, 2680, 1860, 720, 120, 9403, 21234, 30030, 27240, 15480, 5040, 720, 82508, 214459, 358848, 400890, 299040, 143640, 40320, 5040, 808393, 2375736, 4586456, 6077904, 5599440, 3541440, 1471680, 362880, 40320

Offset

0,2

Links

Table of n, a(n) for n=0..44.

Formula

T(n, k) = n * (T(n-1, k-1) + T(n-1, k)) for 0 < k < n with initial values T(n, 0) = A000153(n+1) and T(n, n) = A000142(n).

E.g.f. of column k: (exp(-t) / (1-t)^3) * (t / (1-t))^k.

E.g.f.: exp(x * t / (1-t) - t) / (1-t)^3.

Example

Triangle T(n, k) starts:
n\k :      0       1       2       3       4       5       6      7     8
=========================================================================
  0 :      1
  1 :      2       1
  2 :      7       6       2
  3 :     32      39      24       6
  4 :    181     284     252     120      24
  5 :   1214    2325    2680    1860     720     120
  6 :   9403   21234   30030   27240   15480    5040     720
  7 :  82508  214459  358848  400890  299040  143640   40320   5040
  8 : 808393 2375736 4586456 6077904 5599440 3541440 1471680 362880 40320
  etc.

Prog

(PARI) T(n, k) = { sum(i=0, n-k, (-1)^i * binomial(n-k, i) * (n+2-i)!) * binomial(n, k) / ((k+1) * (k+2)) }

Crossrefs

Cf. A000153 (column 0), A000142 (main diagonal and 1st subdiagonal).

Cf. A000255 (alt. row sums).

Sequence in context: A039814 A178120 A180568 * A248950 A329058 A078301

Adjacent sequences: A373044 A373045 A373049 * A373051 A373052 A373053

Keyword

nonn,easy,tabl

Author

Werner Schulte, May 20 2024

Status

approved