A169654 Triangle T(n, k) = A169643(n, k) - A169653(n, 1) + 1, read by rows.
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
Offset: 1
Examples
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;
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)) >; A169654:= func< n,k | A169653(n, k) - A169653(n, 1) + 1 >; [A169654(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] - (-1)^n*(n! + 1) + 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)) def A169654(n, k): return A169653(n, k) - A169653(n, 1) + 1 flatten([[A169654(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) - 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)
Extensions
Edited by G. C. Greubel, Feb 23 2021