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%I #16 Apr 04 2019 05:16:06
%S 1,2,1,8,2,3,36,8,6,13,172,36,24,26,63,852,172,108,104,126,321,4324,
%T 852,516,468,504,642,1683,22332,4324,2556,2236,2268,2568,3366,8989,
%U 116876,22332,12972,11076,10836,11556,13464,17978,48639
%N Eigentriangle, row sums = A001850, the Delannoy numbers.
%C Row sums = A001850, the Delannoy numbers: (1, 3, 13, 63, 321,...).
%C Sum of n-th row terms = rightmost term of next row.
%H M. Dziemianczuk, <a href="http://www.emis.de/journals/INTEGERS/papers/n54/n54.Abstract.html">Generalizing Delannoy numbers via counting weighted lattice paths</a>, INTEGERS, 13 (2013), #A54.
%H M. Dziemianczuk, <a href="http://arxiv.org/abs/1410.5747">On Directed Lattice Paths With Additional Vertical Steps</a>, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
%H M. Dziemianczuk, <a href="https://doi.org/10.1016/j.disc.2015.11.001">On Directed Lattice Paths With Additional Vertical Steps</a>, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.
%F Triangle read by rows, M*Q. M = an infinite lower triangular matrix with A109980 in every column: (1, 2, 8, 36, 172,...); Q = a matrix with A001850 prefaced with a "1" as the main diagonal: (1, 1, 3, 13, 63, 321,...) and the rest zeros.
%e First few rows of the triangle =
%e 1;
%e 2, 1;
%e 8, 2, 3;
%e 36, 8, 6, 13;
%e 172, 36, 24, 26, 63;
%e 852, 172, 108, 104, 126, 321;
%e 4324, 852, 516, 468, 504, 642, 1683;
%e 22332, 4324, 2556, 2236, 2268, 2568, 3366, 8989;
%e 116876, 22332, 12972, 11076, 10836, 11556, 13464, 17978, 48639;
%e ...
%e Row 3 = (36, 8, 6, 13) = termwise products of (36, 8, 2, 1) and (1, 1, 3, 13).
%t nmax = 8;
%t T[0, 0] = 1;
%t T[n_, 0] := SeriesCoefficient[1/(x + Sqrt[1 - 6x + x^2]), {x, 0, n}];
%t T[n_, n_] := LegendreP[n - 1, 3];
%t row[n_] := row[n] = Table[T[m, 0], {m, n, 0, -1}]*Table[T[m, m], {m, 0, n} ];
%t T[n_, k_] /; 0 < k < n := row[n][[k + 1]];
%t Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 07 2018 *)
%Y Cf. A001850, A109980.
%K eigen,nonn,tabl
%O 0,2
%A _Gary W. Adamson_, Nov 30 2008
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