A174790 Triangle read by rows: T(n,m) = 1 + ((-1 + binomial(n, m))*(n!)^2)/(m!*(n - m)!).

1, 1, 1, 1, 5, 1, 1, 37, 37, 1, 1, 289, 721, 289, 1, 1, 2401, 10801, 10801, 2401, 1, 1, 21601, 151201, 273601, 151201, 21601, 1, 1, 211681, 2116801, 5997601, 5997601, 2116801, 211681, 1, 1, 2257921, 30481921, 124185601, 194745601, 124185601, 30481921, 2257921, 1

Offset

0,5

Links

Stefano Spezia, First 101 rows of the triangle, flattened

Formula

T(n,m) = ((n!)^2/(m!*(n - m)!))*binomial(n, m) - ((n!)^2/(m!*(n - m)!)) + 1.

From Stefano Spezia, Dec 18 2018: (Start)

T(n,m) = 1 + ((-1 + binomial(n, m))*(n!)^2)/(m!*(-m + n)!).

T(n,m) = 1 + ((-1 + A007318(n, m))*A001044(n)/(A000142(m)*A000142(-m + n)).

(End)

Example

Triangle begins:
  1
  1      1
  1      5       1
  1     37      37       1
  1    289     721     289       1
  1   2401   10801   10801    2401       1
  1  21601  151201  273601  151201   21601      1
  ...

Mathematica

T[n_, m_] := (n!^2/(m!(n - m)!))*Binomial[n, m] - (n!^2/(m!(n - m)!)) + 1; Flatten[Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}]]

Prog

(GAP) Flat(List([0..10], n->List([0..n], m->1 + ((- 1 + Binomial(n, m))*(Factorial(n)^2)/(Factorial(m)*Factorial(-m+n))))
)); # Stefano Spezia, Dec 18 2018
(PARI)
T(n, m)= 1 + ((- 1 + binomial(n, m))*(n!)^2)/(m!*(-m+n)!);
tabl(nn) = for(n=0, nn, for(m=0, n, print1(T(n, m), ", ")); print);
tabl(10) \\ Stefano Spezia, Dec 18 2018

Crossrefs

Cf. A007318, A001044, A000142.

Cf. A320329 (row sums).

Sequence in context: A015116 A367380 A322220 * A156691 A246051 A111820

Adjacent sequences: A174787 A174788 A174789 * A174791 A174792 A174793

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 29 2010

Extensions

Edited by Stefano Spezia, Dec 18 2018

Status

approved