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%I #5 Jan 19 2015 04:26:59
%S 0,0,1,0,1,-1,0,1,1,2,0,1,3,-1,-6,0,1,5,2,2,24,0,1,7,11,-2,-6,-120,0,
%T 1,9,26,6,4,24,720,0,1,11,47,50,-6,-12,-120,-5040,0,1,13,74,154,24,12,
%U 48,720,40320,0,1,15,107,342,274,-24,-36,-240,-5040,-362880
%N Square array read by ascending antidiagonals, T(n, k) = k!*[x^k](log(x+1)*sum(j=0..n, C(n,j)*x^j)), n>=0, k>=0.
%F T(n,n) = A000254(n).
%e Square array starts:
%e [n\k][0 1 2 3 4 5 6]
%e [0] 0, 1, -1, 2, -6, 24, -120, ...
%e [1] 0, 1, 1, -1, 2, -6, 24, ...
%e [2] 0, 1, 3, 2, -2, 4, -12, ...
%e [3] 0, 1, 5, 11, 6, -6, 12, ...
%e [4] 0, 1, 7, 26, 50, 24, -24, ...
%e [5] 0, 1, 9, 47, 154, 274, 120, ...
%e [6] 0, 1, 11, 74, 342, 1044, 1764, ...
%e The first few rows as a triangle:
%e 0,
%e 0, 1,
%e 0, 1, -1,
%e 0, 1, 1, 2,
%e 0, 1, 3, -1, -6,
%e 0, 1, 5, 2, 2, 24,
%e 0, 1, 7, 11, -2, -6, -120,
%e 0, 1, 9, 26, 6, 4, 24, 720.
%p T := (n,k) -> k!*coeff(series(ln(x+1)*add(binomial(n,j)*x^j, j=0..n), x, k+1), x, k): for n from 0 to 6 do lprint(seq(T(n,k),k=0..6)) od;
%Y Cf. A000254.
%K sign,tabl
%O 0,10
%A _Peter Luschny_, Jan 18 2015
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