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%I #36 Apr 11 2021 09:53:28
%S 1,2,-1,3,2,-2,4,7,2,-1,5,14,18,2,1,6,23,52,47,2,2,7,34,110,194,123,2,
%T 1,8,47,198,527,724,322,2,-1,9,62,322,1154,2525,2702,843,2,-2,10,79,
%U 488,2207,6726,12098,10084,2207,2,-1,11,98,702,3842,15127,39202,57965,37634,5778,2,1
%N Rectangular array A: first differences of row entries of array A294099, read by antidiagonals.
%C From a problem in A269254. For detailed theory, see [Hone].
%H Andrew N. W. Hone, et al., <a href="https://arxiv.org/abs/1802.01793">On a family of sequences related to Chebyshev polynomials</a>, arXiv:1802.01793 [math.NT], 2018.
%F A(n,k) = T_k(n), n >= 1, k >= 1, where T_j(x) = x*T_{j-1}(x) - T_{j-2}(x), j >= 2, T_0(x) = 2, T_1(x) = x, (dilated Chebyshev polynomials of the first kind).
%e Array begins:
%e 1 -1 -2 -1 1 2 1 -1 -2 -1
%e 2 2 2 2 2 2 2 2 2 2
%e 3 7 18 47 123 322 843 2207 5778 15127
%e 4 14 52 194 724 2702 10084 37634 140452 524174
%e 5 23 110 527 2525 12098 57965 277727 1330670 6375623
%e 6 34 198 1154 6726 39202 228486 1331714 7761798 45239074
%e 7 47 322 2207 15127 103682 710647 4870847 33385282 228826127
%e 8 62 488 3842 30248 238142 1874888 14760962 116212808 914941502
%e 9 79 702 6239 55449 492802 4379769 38925119 345946302 3074591599
%e 10 98 970 9602 95050 940898 9313930 92198402 912670090 9034502498
%t t[n_, 0] := 2; t[n_, 1] := n; t[n_, k_] := n*t[n, k - 1] - t[n, k - 2]; Table[t[n, k], {n, 10}, {k, 10}] // Grid
%Y Main diagonal gives A343259.
%Y Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.
%K sign,tabl
%O 1,2
%A _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Jan 24 2018
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