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%I #24 Jul 27 2024 23:52:47
%S 1,1,4,1,6,18,1,8,32,88,1,10,50,170,450,1,12,72,292,912,2364,1,14,98,
%T 462,1666,4942,12642,1,16,128,688,2816,9424,27008,68464,1,18,162,978,
%U 4482,16722,53154,148626,374274,1,20,200,1340,6800,28004,97880,299660,822560,2060980,1,22,242,1782,9922,44726,170610,568150,1690370,4573910,11414898
%N T(n,k) is the coordination number of the (n+1)-dimensional cubic lattice for radius k; triangle read by rows, n>=0, 0<=k<=n.
%H J. Schroder, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Schroder/schroder45.html">Generalized Schroder Numbers and the Rotation principle</a>, J. Int. Seq. 10 (2007) # 07.7.7, Theorem 4.2.
%F G.f.: (1+y)/(1-x-y-x*y).
%F T(n,k) = A008288(n,k) + A008288(n,k-1).
%e The full array starts
%e 1 2 2 2 2 2 2 2 2
%e 1 4 8 12 16 20 24 28 32
%e 1 6 18 38 66 102 146 198 258
%e 1 8 32 88 192 360 608 952 1408
%e 1 10 50 170 450 1002 1970 3530 5890
%e 1 12 72 292 912 2364 5336 10836 20256
%e 1 14 98 462 1666 4942 12642 28814 59906
%e 1 16 128 688 2816 9424 27008 68464 157184
%e 1 18 162 978 4482 16722 53154 148626 374274
%p A343599 := proc(n,k)
%p local g,x,y ;
%p g := (1+y)/(1-x-y-x*y) ;
%p coeftayl(%,x=0,n) ;
%p coeftayl(%,y=0,k) ;
%p end proc:
%t T[n_, k_] := Module[{x, y}, SeriesCoefficient[(1 + y)/(1 - x - y - x*y), {x, 0, n}] // SeriesCoefficient[#, {y, 0, k}]&];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 16 2023 *)
%Y Cf. A035607 (by antidiags), A008574 (n=1), A005899 (n=2), A008412 (n=3), A008413 (n=4), A008414 (n=5), A001105 (k=2), A035597 (k=3), A035598 (k=4).
%Y Main diagonal gives A050146(n+1).
%K nonn,tabl,easy
%O 0,3
%A _R. J. Mathar_, Apr 21 2021