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A174689
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Triangle T(n, k) = n! * binomial(n, k)^2 - n! + 1, read by rows.
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1
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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
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OFFSET
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0,5
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LINKS
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FORMULA
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T(n, k) = n! * binomial(n, k)^2 - n! + 1.
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)
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EXAMPLE
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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;
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MATHEMATICA
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T[n_, k_]:= n!*Binomial[n, k]^2 - n! + 1;
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
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PROG
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(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
(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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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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