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%I #34 Sep 26 2019 03:42:16
%S 0,0,1,0,1,3,0,1,4,4,0,1,5,8,7,0,1,6,13,15,11,0,1,7,19,28,26,18,0,1,8,
%T 26,47,54,44,29,0,1,9,34,73,101,98,73,47,0,1,10,43,107,174,199,171,
%U 120,76,0,1,11,53,150,281,373,370,291,196,123
%N Hyperlucas number array T(r,n) = L(n)^(r), read by ascending antidiagonals (r >= 0, n >= 0).
%C In Theorem 17, Dil and Mezo (2008) connect the hyperlucas numbers (this array) with the incomplete Lucas numbers (A324242). - _Petros Hadjicostas_, Sep 03 2019
%H Ayhan Dil and Istvan Mezo, <a href="http://arXiv.org/abs/0803.4388">A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers</a>, arXiv:0803.4388 [math.NT], 2008.
%H Ayhan Dil and Istvan Mezo, <a href="https://doi.org/10.1016/j.amc.2008.10.013">A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers</a>, Applied Mathematics and Computation 206(2) (2008), 942-951.
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/SteenrodAlgebra.html">Steenrod Algebra</a>.
%F T(r,n) = L(n)^(r) = Apply partial sum operator r times to Lucas numbers A000204.
%F From _Petros Hadjicostas_, Sep 03 2019: (Start)
%F T(r, n) = L(n)^(r) = Sum_{k = 0..n} L(k)^(r-1) for r >= 1, with T(0,n) = L(n)^(0) = L(n) = A000204(n), T(r,0) = L(0)^(r) = 0, and T(r,1) = L(1)^(r) = 1. (See Definition 13 in Dil and Mezo (2008).)
%F G.f. for row r: Sum_{n >= 0} L(n)^(r)*t^n = t * (1+2*t)/((1-t-t^2) * (1-t)^r). (Corrected from Proposition 14 in Dil and Mezo (2008).)
%F (End)
%e The array T(r,n) = L(n)^(r) begins:
%e .....|n=0|n=1|.n=2|.n=3|.n=4.|.n=5.|..n=6.|.n=7..|..n=8..|..n=9..|.n=10..|.in.OEIS
%e r=0..|.0.|.1.|..3.|..4.|...7.|..11.|...18.|...29.|....47.|....76.|...123.|.A000204
%e r=1..|.0.|.1.|..4.|..8.|..15.|..26.|...44.|...73.|...120.|...196.|...319.|.A027961
%e r=2..|.0.|.1.|..5.|.13.|..28.|..54.|...98.|..171.|...291.|...487.|...806.|.A023537
%e r=3..|.0.|.1.|..6.|.19.|..47.|.101.|..199.|..370.|...661.|..1148.|..1954.|.A027963
%e r=4..|.0.|.1.|..7.|.26.|..73.|.174.|..373.|..743.|..1404.|..2552.|..4506.|.A027964
%e r=5..|.0.|.1.|..8.|.34.|.107.|.281.|..654.|.1397.|..2801.|..5353.|..9859.|.A053298
%e r=6..|.0.|.1.|..9.|.43.|.150.|.431.|.1085.|.2482.|..5283.|.10636.|.20495.|.new
%e r=7..|.0.|.1.|.10.|.53.|.203.|.634.|.1719.|.4201.|..9484.|.20120.|.40615.|.new
%e r=8..|.0.|.1.|.11.|.64.|.267.|.901.|.2620.|.6821.|.16305.|.36425.|.77040.|.new
%e r=9..|.0.|.1.|.12.|.76.|.343.|1244.|.3864.|10685.|.26990.|.63415.|140455.|.new
%e For example, T(4,5) = L(5)^(4) = L(0)^(3) + L(1)^(3) + L(2)^(3) + L(3)^(3) + L(4)^(3) + L(5)^(3) = 0 + 1 + 6 + 19 + 47 + 101 = 174. - _Petros Hadjicostas_, Sep 03 2019
%p L:= proc(r, n) option remember; `if`(n=0, 0, `if`(r=0,
%p `if`(n<3, 2*n-1, L(0, n-2)+L(0, n-1)), L(r-1, n)+L(r, n-1)))
%p end:
%p seq(seq(L(d-n, n), n=0..d), d=0..12); # _Alois P. Heinz_, Sep 03 2019
%t L[r_, n_] := L[r, n] = If[n == 0, 0, If[r == 0, If[n < 3, 2n-1, L[0, n-2] + L[0, n-1]], L[r-1, n] + L[r, n-1]]];
%t Table[L[d-n, n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Sep 26 2019, from Maple *)
%Y Cf. A000204, A027961, A023537, A027963, A027964, A053298, A123736, A136431.
%Y Cf. A038730, A038792, and A134511 for incomplete Fibonacci sequences, and A324242 for incomplete Lucas sequences.
%K easy,nonn,tabl
%O 0,6
%A _Jonathan Vos Post_, Apr 04 2008
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