A177848 Triangle, read by rows, T(n, k) = t(k, n-k+1) - t(1, n) + 1 where t(n, m) = (n*m)!*Beta(n, m).

1, 1, 1, 1, 3, 1, 1, 55, 55, 1, 1, 1993, 12073, 1993, 1, 1, 120841, 7983241, 7983241, 120841, 1, 1, 11404081, 12454040881, 149448498481, 12454040881, 11404081, 1, 1, 1556750161, 38109367290961, 8688935743482961, 8688935743482961, 38109367290961, 1556750161, 1

Offset

1,5

Comments

Row sums are {1, 2, 5, 112, 16061, 16208166, 174379388407, 17454093335048168, 27083470639271574245769, 421762213493139881153379087370, ...}.

Links

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Formula

Let t(n, k) = (n*k)!*Beta(n, k) then T(n, k) = t(k, n-k+1) - t(1, n) + 1.

Example

Triangle begins as:
  1;
  1,        1;
  1,        3,           1;
  1,       55,          55,            1;
  1,     1993,       12073,         1993,           1;
  1,   120841,     7983241,      7983241,      120841,        1;
  1, 11404081, 12454040881, 149448498481, 12454040881, 11404081, 1;

Mathematica

t[n_, k_]:= (n*k)!*Beta[n, k];
Table[t[k, n-k+1] - t[1, n] + 1, {n, 12}, {k, n}]//Flatten

Prog

(Sage)
def t(n, k): return factorial(n*k)*beta(n, k)
flatten([[t(k, n-k+1) - t(1, n) + 1 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 06 2021

Crossrefs

Cf. A060854.

Sequence in context: A173505 A253178 A174587 * A168549 A010272 A368026

Adjacent sequences: A177845 A177846 A177847 * A177849 A177850 A177851

Keyword

nonn,tabl

Author

Roger L. Bagula, May 14 2010

Extensions

Edited by G. C. Greubel, Feb 06 2021

Status

approved