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%I #10 Jul 26 2015 10:01:00
%S 1,-1,6,2,-10,10,-17,84,-70,28,124,-612,504,-168,36,-2764,13640,
%T -11220,3696,-660,88,43688,-215592,177320,-58344,10296,-1144,104,
%U -1859138,9174480,-7545720,2482480,-437580,48048,-3640,240,51236656,-252842768,207954880,-68414528,12057760,-1322464,99008,-5440
%N Triangle T(n,k) read by rows: see formula lines for definition.
%D H. W. Gould, Power sum identities for arbitrary symmetric arrays, SIAM J. Appl. Math., 17 (1969), 307-316.
%F T(n, n) = (2n+1)2^floor((n+1)/2), n >= 0.
%F 2^-floor((n+2)/2)*T(n+1, k) = binomial(2n+3, 2k) - Sum_{j=k..n} binomial(2n+3, 2j+1)*2^-floor((j+3)/2)*T(j, k), k=0..n.
%e Triangle begins:
%e 1
%e -1 6
%e 2 -10 10
%e -17 84 -70 28
%e 124 -612 504 -168 36
%t T[n_, n_] := (2n + 1)2^Floor[(n + 1)/2]; T[n_, k_] := (Binomial[2n + 1, 2k] - Sum[ Binomial[2n + 1, 2j + 1]*2^-Floor[(j + 3)/2]*T[j, k], {j, k, n - 1}])(2^Floor[(n + 1)/2]); Flatten[ Table[ T[n, k], {n, 0, 8}, {k, 0, n}]] (* _Robert G. Wilson v_, May 10 2005 *)
%Y Cf. A097578, A097716, A097749.
%K sign,tabl,easy
%O 0,3
%A _N. J. A. Sloane_, Sep 21 2004
%E More terms from _Emeric Deutsch_, Dec 24 2004