A174689 Triangle T(n, k) = n! * binomial(n, k)^2 - n! + 1, read by rows.
1, 1, 1, 1, 7, 1, 1, 49, 49, 1, 1, 361, 841, 361, 1, 1, 2881, 11881, 11881, 2881, 1, 1, 25201, 161281, 287281, 161281, 25201, 1, 1, 241921, 2217601, 6168961, 6168961, 2217601, 241921, 1, 1, 2540161, 31570561, 126403201, 197527681, 126403201, 31570561, 2540161, 1
Offset: 0
Examples
Triangle begins as: 1; 1, 1; 1, 7, 1; 1, 49, 49, 1; 1, 361, 841, 361, 1; 1, 2881, 11881, 11881, 2881, 1; 1, 25201, 161281, 287281, 161281, 25201, 1; 1, 241921, 2217601, 6168961, 6168961, 2217601, 241921, 1; 1, 2540161, 31570561, 126403201, 197527681, 126403201, 31570561, 2540161, 1;
Links
- G. C. Greubel, Rows n = 0..100 of the triangle, flattened
Programs
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Magma
[Factorial(n)*(Binomial(n, k)^2 -1) + 1: k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 10 2021
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Mathematica
T[n_, k_]:= n!*Binomial[n, k]^2 - n! + 1; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten -
Sage
flatten([[factorial(n)*(binomial(n, k)^2 -1) + 1 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 10 2021
Formula
T(n, k) = n! * binomial(n, k)^2 - n! + 1.
From G. C. Greubel, Feb 10 2021: (Start)
T(n, k) = n! * ( A008459(n, k) - 1 ) + 1.
Sum_{k=0..n} T(n, k) = (n+1)*( n!*( C_{n} - 1 ) + 1 ) = (n+1)*( n!*( A000108(n) - 1 ) + 1). (End)
Extensions
Edited by G. C. Greubel, Feb 10 2021