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A169653 Triangle T(n,k) = A008297(n,k) + A008297(n,n-k+1), read by rows. 2
-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, -362881, -1451592, -1695456, -874944, -423360, -874944, -1695456, -1451592, -362881 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
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).
T(n, k) = A008297(n,k) + A008297(n,n-k+1).
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (-1)^n * (A105278(n, k) + A105278(n, n-k+1)).
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
def A169653(n, k): return (-1)^n*A001263(n, k)*(factorial(k) + factorial(n-k+1))
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:= func< n, k | (-1)^n*A001263(n, k)*(Factorial(k) + Factorial(n-k+1)) >;
[A169653(n, k): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 23 2021
CROSSREFS
Sequence in context: A095978 A370361 A156763 * A129012 A136122 A121875
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Apr 05 2010
EXTENSIONS
Edited by G. C. Greubel, Feb 23 2021
STATUS
approved

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Last modified July 18 13:34 EDT 2024. Contains 374378 sequences. (Running on oeis4.)