A169658 Triangle, read by rows, defined by T(n, k) = b(n, k) + b(n, n-k+1) - (b(n,1) + b(n,n)) + 1, where b(n, k) = (-1)^n*(n!/k!)^2 *binomial(n-1, k-1).
1, 1, 1, 1, 2, 1, 1, -96, -96, 1, 1, -98, 9602, -98, 1, 1, 129780, -365400, -365400, 129780, 1, 1, -12701092, 14791142, 23637602, 14791142, -12701092, 1, 1, 1219277248, -677310144, -1522967040, -1522967040, -677310144, 1219277248, 1
Offset: 1
Examples
Triangle begins as: 1; 1, 1; 1, 2, 1; 1, -96, -96, 1; 1, -98, 9602, -98, 1; 1, 129780, -365400, -365400, 129780, 1; 1, -12701092, 14791142, 23637602, 14791142, -12701092, 1;
Links
- G. C. Greubel, Rows n = 1..100 of triangle, flattened
Crossrefs
Cf. A008297.
Programs
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Magma
b:= func< n,k | (-1)^n*(Factorial(n)/Factorial(k))^2*Binomial(n-1, k-1) >; [[b(n, k) +b(n, n-k+1) -b(n,1) -b(n,n) +1: k in [1..n]]: n in [1..10]]; // G. C. Greubel, May 20 2019
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Mathematica
L[n_, m_] = (-1)^n*(n!/m!)^2*Binomial[n-1, m-1]; t[n_, m_] = L[n, m] + L[n, n-m+1]; Table[t[n, m] - t[n, 1] + 1, {n, 1, 10}, {m, 1, n}]//Flatten -
PARI
b(n, k) = (-1)^n*(n!/k!)^2 *binomial(n-1, k-1); t(n, k) = b(n, k) + b(n, n-k+1); for(n=1, 10, for(k=1, n, print1(t(n,k) - t(n,1) + 1, ", "))) \\ G. C. Greubel, May 20 2019
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Sage
def b(n, k): return (-1)^n*factorial(n-k)^2*binomial(n,k)^2*binomial(n-1, k-1) def t(n, k): return b(n, k) + b(n, n-k+1) [[t(n,k) - t(n,1) + 1 for k in (1..n)] for n in (1..10)] # G. C. Greubel, May 20 2019
Formula
T(n, k) = b(n, k) + b(n, n-k+1) - b(n, n) - b(n, 1) + 1, where b(n, k) = (-1)^n*(n!/m!)^2 *binomial(n-1, k-1), where 1 <= k <= n, n >= 1.
Extensions
Edited by G. C. Greubel, May 20 2019
Comments