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Triangle T(n, k) = n!*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - n! + 1, read by rows.
2

%I #8 Feb 10 2021 01:38:03

%S 1,1,1,1,13,1,1,121,121,1,1,1081,2281,1081,1,1,10081,35281,35281,

%T 10081,1,1,100801,524161,876961,524161,100801,1,1,1088641,7862401,

%U 19716481,19716481,7862401,1088641,1,1,12700801,121564801,426384001,639757441,426384001,121564801,12700801,1

%N Triangle T(n, k) = n!*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - n! + 1, read by rows.

%H G. C. Greubel, <a href="/A174694/b174694.txt">Rows n = 1..100 of the triangle, flattened</a>

%F T(n, k) = n!*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - n! + 1.

%F From _G. C. Greubel_, Feb 09 2021: (Start)

%F T(n, k) = (-1)^n * k! * A176013(n, k) - n! + 1.

%F Sum_{k=1..n} T(n,k) = n! * (C_{n} - n) + n, where C_{n} are the Catalan numbers (A000108). (End)

%e Triangle begin as:

%e 1;

%e 1, 1;

%e 1, 13, 1;

%e 1, 121, 121, 1;

%e 1, 1081, 2281, 1081, 1;

%e 1, 10081, 35281, 35281, 10081, 1;

%e 1, 100801, 524161, 876961, 524161, 100801, 1;

%e 1, 1088641, 7862401, 19716481, 19716481, 7862401, 1088641, 1;

%e 1, 12700801, 121564801, 426384001, 639757441, 426384001, 121564801, 12700801, 1;

%t T[n_, k_]:= n!*(1/k)*Binomial[n-1, k-1]*Binomial[n, k-1] - n! + 1;

%t Table[T[n, k], {n,12}, {k,n}]//Flatten

%o (Sage)

%o def A174694(n, k): return (factorial(n)/k)*binomial(n-1, k-1)*binomial(n, k-1) - factorial(n) + 1

%o flatten([[A174694(n, k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Feb 09 2021

%o (Magma)

%o A174694:= func< n, k | (Factorial(n)/k)*Binomial(n-1, k-1)*Binomial(n, k-1) - Factorial(n) + 1 >;

%o [A174694(n, k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Feb 09 2021

%Y Cf. A000108, A174696, A176013.

%K nonn,tabl,easy

%O 1,5

%A _Roger L. Bagula_, Mar 27 2010

%E Edited by _G. C. Greubel_, Feb 09 2021