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Square array read by ascending antidiagonals, T(n, k) = k!*[x^k](log(x+1)*sum(j=0..n, C(2*n,j)*x^j)), n>=0, k>=0.
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%I #5 Jan 19 2015 04:27:14

%S 0,0,1,0,1,-1,0,1,3,2,0,1,7,-4,-6,0,1,11,26,10,24,0,1,15,74,-46,-36,

%T -120,0,1,19,146,342,144,168,720,0,1,23,242,1066,-756,-624,-960,-5040,

%U 0,1,27,362,2414,5944,2844,3408,6480,40320,0,1,31,506,4578,19524

%N Square array read by ascending antidiagonals, T(n, k) = k!*[x^k](log(x+1)*sum(j=0..n, C(2*n,j)*x^j)), n>=0, k>=0.

%F T(n,n) = A098118(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, 3, -4, 10, -36, 168, ...

%e [2] 0, 1, 7, 26, -46, 144, -624, ...

%e [3] 0, 1, 11, 74, 342, -756, 2844, ...

%e [4] 0, 1, 15, 146, 1066, 5944, -15768, ...

%e [5] 0, 1, 19, 242, 2414, 19524, 127860, ...

%e [6] 0, 1, 23, 362, 4578, 48504, 434568, ...

%e The first few rows as a triangle:

%e 0,

%e 0, 1,

%e 0, 1, -1,

%e 0, 1, 3, 2,

%e 0, 1, 7, -4, -6,

%e 0, 1, 11, 26, 10, 24,

%e 0, 1, 15, 74, -46, -36, -120,

%e 0, 1, 19, 146, 342, 144, 168, 720.

%p T := (n,k) -> k!*coeff(series(ln(x+1)*add(binomial(2*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. A098118.

%K sign,tabl

%O 0,9

%A _Peter Luschny_, Jan 18 2015