A169653 Triangle T(n,k) = A008297(n,k) + A008297(n,n-k+1), read by rows.
-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
Offset: 1
Examples
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;
Links
- G. C. Greubel, Rows n = 1..100 of the triangle, flattened
Programs
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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
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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 *) -
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
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)
Extensions
Edited by G. C. Greubel, Feb 23 2021