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A174694 Triangle T(n, k) = n!*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - n! + 1, read by rows.

%I A174694 #8 Feb 10 2021 01:38:03
%S A174694 1,1,1,1,13,1,1,121,121,1,1,1081,2281,1081,1,1,10081,35281,35281,
%T A174694 10081,1,1,100801,524161,876961,524161,100801,1,1,1088641,7862401,
%U A174694 19716481,19716481,7862401,1088641,1,1,12700801,121564801,426384001,639757441,426384001,121564801,12700801,1
%N A174694 Triangle T(n, k) = n!*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - n! + 1, read by rows.
%H A174694 G. C. Greubel, <a href="/A174694/b174694.txt">Rows n = 1..100 of the triangle, flattened</a>
%F A174694 T(n, k) = n!*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - n! + 1.
%F A174694 From _G. C. Greubel_, Feb 09 2021: (Start)
%F A174694 T(n, k) = (-1)^n * k! * A176013(n, k) - n! + 1.
%F A174694 Sum_{k=1..n} T(n,k) = n! * (C_{n} - n) + n, where C_{n} are the Catalan numbers (A000108). (End)
%e A174694 Triangle begin as:
%e A174694   1;
%e A174694   1,        1;
%e A174694   1,       13,         1;
%e A174694   1,      121,       121,         1;
%e A174694   1,     1081,      2281,      1081,         1;
%e A174694   1,    10081,     35281,     35281,     10081,         1;
%e A174694   1,   100801,    524161,    876961,    524161,    100801,         1;
%e A174694   1,  1088641,   7862401,  19716481,  19716481,   7862401,   1088641,        1;
%e A174694   1, 12700801, 121564801, 426384001, 639757441, 426384001, 121564801, 12700801, 1;
%t A174694 T[n_, k_]:= n!*(1/k)*Binomial[n-1, k-1]*Binomial[n, k-1] - n! + 1;
%t A174694 Table[T[n, k], {n,12}, {k,n}]//Flatten
%o A174694 (Sage)
%o A174694 def A174694(n, k): return (factorial(n)/k)*binomial(n-1, k-1)*binomial(n, k-1) - factorial(n) + 1
%o A174694 flatten([[A174694(n, k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Feb 09 2021
%o A174694 (Magma)
%o A174694 A174694:= func< n, k | (Factorial(n)/k)*Binomial(n-1, k-1)*Binomial(n, k-1) - Factorial(n) + 1 >;
%o A174694 [A174694(n, k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Feb 09 2021
%Y A174694 Cf. A000108, A174696, A176013.
%K A174694 nonn,tabl,easy
%O A174694 1,5
%A A174694 _Roger L. Bagula_, Mar 27 2010
%E A174694 Edited by _G. C. Greubel_, Feb 09 2021