OFFSET
1,1
LINKS
G. C. Greubel, Rows n = 1..100 of the triangle, flattened
FORMULA
T(n, k) = t(n, k) + t(n, n-k+1), where t(n, k) = (-1)^n*(n!/k!)*binomial(n-1, k-1).
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (-1)^n * ( k! + (n-k+1)! ) * A001263(n, k).
Sum_{k=1..n} T(n, k) = 2 * (-1)^n * A000262(n). (End)
EXAMPLE
Triangle begins as:
-2;
3, 3;
-7, -12, -7;
25, 48, 48, 25;
-121, -260, -240, -260, -121;
721, 1830, 1500, 1500, 1830, 721;
-5041, -15162, -13230, -8400, -13230, -15162, -5041;
40321, 141176, 142296, 70560, 70560, 142296, 141176, 40321;
MATHEMATICA
t[n_, m_] = (-1)^n*(n!/m!)*Binomial[n-1, m-1];
T[n_, m_] = t[n, m] + t[n, n-m+1];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Feb 23 2021 *)
PROG
(Sage)
def A001263(n, k): return binomial(n-1, k-1)*binomial(n, k-1)/k
flatten([[A169653(n, k) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Feb 23 2021
(Magma)
A001263:= func< n, k | Binomial(n-1, k-1)*Binomial(n, k-1)/k >;
[A169653(n, k): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 23 2021
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Apr 05 2010
EXTENSIONS
Edited by G. C. Greubel, Feb 23 2021
STATUS
approved