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%I #7 Feb 06 2021 22:31:16
%S 1,1,1,1,3,1,1,55,55,1,1,1993,12073,1993,1,1,120841,7983241,7983241,
%T 120841,1,1,11404081,12454040881,149448498481,12454040881,11404081,1,
%U 1,1556750161,38109367290961,8688935743482961,8688935743482961,38109367290961,1556750161,1
%N Triangle, read by rows, T(n, k) = t(k, n-k+1) - t(1, n) + 1 where t(n, m) = (n*m)!*Beta(n, m).
%C Row sums are {1, 2, 5, 112, 16061, 16208166, 174379388407, 17454093335048168, 27083470639271574245769, 421762213493139881153379087370, ...}.
%H G. C. Greubel, <a href="/A177848/b177848.txt">Rows n = 1..30 of the triangle, flattened</a>
%F Let t(n, k) = (n*k)!*Beta(n, k) then T(n, k) = t(k, n-k+1) - t(1, n) + 1.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 55, 55, 1;
%e 1, 1993, 12073, 1993, 1;
%e 1, 120841, 7983241, 7983241, 120841, 1;
%e 1, 11404081, 12454040881, 149448498481, 12454040881, 11404081, 1;
%t t[n_, k_]:= (n*k)!*Beta[n, k];
%t Table[t[k, n-k+1] - t[1, n] + 1, {n, 12}, {k, n}]//Flatten
%o (Sage)
%o def t(n, k): return factorial(n*k)*beta(n, k)
%o flatten([[t(k, n-k+1) - t(1,n) + 1 for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Feb 06 2021
%Y Cf. A060854.
%K nonn,tabl
%O 1,5
%A _Roger L. Bagula_, May 14 2010
%E Edited by _G. C. Greubel_, Feb 06 2021
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