%I #10 Jul 02 2013 09:30:50

%S 1,-1,1,1,-3,2,3,3,-12,6,-2,30,8,-60,24,-240,240,240,0,-360,120,-3900,

%T -540,4800,1800,-360,-2520,720,-15120,-112560,65520,70560,12600,-5880,

%U -20160,5040,2169888,-4284000,-756672,2076480,945504,70560,-80640,-181440,40320

%N Triangle read by rows: T(n,k) = n!*S(n,k), where S(n,k) is the matrix inverse of the triangle zeta(k-n,1) - zeta(k-n,k+1), n>=1, k>=1.

%D J. Faulhaber, Academia Algebrae, Darinnen die miraculosische inventiones zu den höchsten Cossen weiters continuirt und profitirt werden, Augspurg, bey Johann Ulrich Schönigs, 1631.

%e 1,

%e -1, 1,

%e 1, -3, 2,

%e 3, 3, -12, 6,

%e -2, 30, 8, -60, 24,

%e -240, 240, 240, 0, -360, 120,

%e -3900, -540, 4800, 1800, -360, -2520, 720.

%p with(linalg): S := proc(n) f := (n,k) -> `if`(k>n,0,Zeta(0,k-n,1)-Zeta(0,k-n,k+1)); inverse(matrix(n,n,f)) end: A214435_row := n -> n!*convert(row(S(n),n),list); for n from 1 to 9 do A214435_row(n) od;

%t max = 9; s = Table[ If[ k > n, 0, Zeta[k - n, 1] - Zeta[k - n, k + 1]], {n, 1, max}, {k, 1, max}] // Inverse; t[n_, k_] := n!*s[[n, k]]; Table[t[n, k], {n, 1, max}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 02 2013 *)

%Y Cf. A103438.

%K sign,tabl

%O 1,5

%A _Peter Luschny_, Jul 17 2012