A097474 Triangle T(n,k) read by rows: see formula lines for definition.
1, -1, 6, 2, -10, 10, -17, 84, -70, 28, 124, -612, 504, -168, 36, -2764, 13640, -11220, 3696, -660, 88, 43688, -215592, 177320, -58344, 10296, -1144, 104, -1859138, 9174480, -7545720, 2482480, -437580, 48048, -3640, 240, 51236656, -252842768, 207954880, -68414528, 12057760, -1322464, 99008, -5440
Offset: 0
Examples
Triangle begins: 1 -1 6 2 -10 10 -17 84 -70 28 124 -612 504 -168 36
References
- H. W. Gould, Power sum identities for arbitrary symmetric arrays, SIAM J. Appl. Math., 17 (1969), 307-316.
Programs
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Mathematica
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 *)
Formula
T(n, n) = (2n+1)2^floor((n+1)/2), n >= 0.
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.
Extensions
More terms from Emeric Deutsch, Dec 24 2004