OFFSET
1,5
LINKS
G. C. Greubel, Rows n = 1..100 of the triangle, flattened
FORMULA
T(n, k) = n!*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - n! + 1.
From G. C. Greubel, Feb 09 2021: (Start)
T(n, k) = (-1)^n * k! * A176013(n, k) - n! + 1.
Sum_{k=1..n} T(n,k) = n! * (C_{n} - n) + n, where C_{n} are the Catalan numbers (A000108). (End)
EXAMPLE
Triangle begin as:
1;
1, 1;
1, 13, 1;
1, 121, 121, 1;
1, 1081, 2281, 1081, 1;
1, 10081, 35281, 35281, 10081, 1;
1, 100801, 524161, 876961, 524161, 100801, 1;
1, 1088641, 7862401, 19716481, 19716481, 7862401, 1088641, 1;
1, 12700801, 121564801, 426384001, 639757441, 426384001, 121564801, 12700801, 1;
MATHEMATICA
T[n_, k_]:= n!*(1/k)*Binomial[n-1, k-1]*Binomial[n, k-1] - n! + 1;
Table[T[n, k], {n, 12}, {k, n}]//Flatten
PROG
(Sage)
def A174694(n, k): return (factorial(n)/k)*binomial(n-1, k-1)*binomial(n, k-1) - factorial(n) + 1
flatten([[A174694(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 09 2021
(Magma)
A174694:= func< n, k | (Factorial(n)/k)*Binomial(n-1, k-1)*Binomial(n, k-1) - Factorial(n) + 1 >;
[A174694(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 09 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Mar 27 2010
EXTENSIONS
Edited by G. C. Greubel, Feb 09 2021
STATUS
approved