A174696 Triangle T(n, k) = n!*(1/k)^2*(binomial(n-1, k-1)*binomial(n, k-1))^2 - n! + 1, read by rows.

1, 1, 1, 1, 49, 1, 1, 841, 841, 1, 1, 11881, 47881, 11881, 1, 1, 161281, 1799281, 1799281, 161281, 1, 1, 2217601, 55560961, 154344961, 55560961, 2217601, 1, 1, 31570561, 1548892801, 9680791681, 9680791681, 1548892801, 31570561, 1, 1, 469929601, 40967337601, 501853968001, 1129171881601, 501853968001, 40967337601, 469929601, 1

Offset: 1

Author

Roger L. Bagula, Mar 27 2010

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins as:
  1;
  1,        1;
  1,       49,          1;
  1,      841,        841,          1;
  1,    11881,      47881,      11881,          1;
  1,   161281,    1799281,    1799281,     161281,          1;
  1,  2217601,   55560961,  154344961,   55560961,    2217601,        1;
  1, 31570561, 1548892801, 9680791681, 9680791681, 1548892801, 31570561, 1;
		

Links

• G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Crossrefs

Cf. A174158, A174694, A176013, A319743.

Programs

Magma

A174696:= func< n, k | (Factorial(n)/k^2)*(Binomial(n-1, k-1)*Binomial(n, k-1))^2 - Factorial(n) + 1 >;
[A174696(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 09 2021

Mathematica

T[n_, m_]:= n!*(1/k)^2*(Binomial[n-1, k-1]*Binomial[n, k-1])^2 - n! + 1;
Table[T[n, k], {n,12}, {k,n}]//Flatten

Sage

def A174696(n, k): return (factorial(n)/k^2)*(binomial(n-1, k-1)*binomial(n, k-1))^2 - factorial(n) + 1
flatten([[A174696(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 09 2021

Formula

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

From G. C. Greubel, Feb 09 2021: (Start)

T(n, k) = n! * A174158(n, k) - n! + 1.

Sum_{k=1..n} T(n,k) = n! * Hypergeometric4F3([-n, -n, 1-n, 1-n], [1, 2, 2], 1) - n*(n! - 1) = n! * A319743(n) - n*(n! - 1). (End)

Extensions

Edited by G. C. Greubel, Feb 09 2021