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A343599 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.

%I A343599 #24 Jul 27 2024 23:52:47
%S A343599 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 A343599 462,1666,4942,12642,1,16,128,688,2816,9424,27008,68464,1,18,162,978,
%U A343599 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 A343599 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 A343599 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 A343599 G.f.: (1+y)/(1-x-y-x*y).
%F A343599 T(n,k) = A008288(n,k) + A008288(n,k-1).
%e A343599 The full array starts
%e A343599      1      2      2      2      2      2      2      2      2
%e A343599      1      4      8     12     16     20     24     28     32
%e A343599      1      6     18     38     66    102    146    198    258
%e A343599      1      8     32     88    192    360    608    952   1408
%e A343599      1     10     50    170    450   1002   1970   3530   5890
%e A343599      1     12     72    292    912   2364   5336  10836  20256
%e A343599      1     14     98    462   1666   4942  12642  28814  59906
%e A343599      1     16    128    688   2816   9424  27008  68464 157184
%e A343599      1     18    162    978   4482  16722  53154 148626 374274
%p A343599 A343599 := proc(n,k)
%p A343599     local g,x,y ;
%p A343599     g := (1+y)/(1-x-y-x*y) ;
%p A343599     coeftayl(%,x=0,n) ;
%p A343599     coeftayl(%,y=0,k) ;
%p A343599 end proc:
%t A343599 T[n_, k_] := Module[{x, y}, SeriesCoefficient[(1 + y)/(1 - x - y - x*y), {x, 0, n}] // SeriesCoefficient[#, {y, 0, k}]&];
%t A343599 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 16 2023 *)
%Y A343599 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 A343599 Main diagonal gives A050146(n+1).
%K A343599 nonn,tabl,easy
%O A343599 0,3
%A A343599 _R. J. Mathar_, Apr 21 2021