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A174696
Triangle T(n, k) = n!*(1/k)^2*(binomial(n-1, k-1)*binomial(n, k-1))^2 - n! + 1, read by rows.
3
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
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
KEYWORD
nonn,tabl,easy
AUTHOR
Roger L. Bagula, Mar 27 2010
EXTENSIONS
Edited by G. C. Greubel, Feb 09 2021
STATUS
approved