OFFSET
1,5
LINKS
G. C. Greubel, Rows n = 1..100 of the triangle, flattened
FORMULA
T(n, k) = t(n, k) + t(n, n-k+1) - t(n, 1) - t(n, n) + 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) = A169653(n, k) - (-1)^n * (n! + 1) + 1.
Sum_{k=1..n} T(n, k) = (-1)^n *(2 * A000262(n) - n*(n! + 1) + (-1)^n * n). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, -4, 1;
1, 24, 24, 1;
1, -138, -118, -138, 1;
1, 1110, 780, 780, 1110, 1;
1, -10120, -8188, -3358, -8188, -10120, 1;
1, 100856, 101976, 30240, 30240, 101976, 100856, 1;
1, -1088710, -1332574, -512062, -60478, -512062, -1332574, -1088710, 1;
1, 12700890, 18147240, 9132480, 816480, 816480, 9132480, 18147240, 12700890, 1;
MATHEMATICA
t[n_, m_] = (-1)^n*(n!/m!)*Binomial[n-1, m-1];
T[n_, m_] = t[n, m] + t[n, n-m+1] - (-1)^n*(n! + 1) + 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([[A169654(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 >;
[A169654(n, k): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 23 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Apr 05 2010
EXTENSIONS
Edited by G. C. Greubel, Feb 23 2021
STATUS
approved