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

1,5

Links

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

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)

Example

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;

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

Prog

(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
(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

Crossrefs

Cf. A174158, A174694, A176013, A319743.

Sequence in context: A115480 A214953 A005071 * A203506 A254617 A012120

Adjacent sequences: A174693 A174694 A174695 * A174697 A174698 A174699

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 27 2010

Extensions

Edited by G. C. Greubel, Feb 09 2021

Status

approved