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A353091 Irregular triangle read by rows: T(n, k) is the number of n-step closed walks on the hexagonal lattice having algebraic area k.

%I A353091 #8 Apr 23 2022 09:43:41
%S A353091 1,6,0,6,66,0,12,0,150,0,30,1020,0,420,0,84,0,6,0,3444,0,1302,0,252,0,
%T A353091 42,19890,0,11952,0,4284,0,984,0,216,0,24,0,82062,0,42972,0,14814,0,
%U A353091 4248,0,990,0,216,0,18,449976,0,327420,0,158970,0,57180,0,18780,0,5190,0,1350,0,270,0,30
%N A353091 Irregular triangle read by rows: T(n, k) is the number of n-step closed walks on the hexagonal lattice having algebraic area k.
%C A353091 Rows 0 and 2 have 1 element each; row 1 is empty; for n > 2, we have 0 <= k <= A069813(n).
%C A353091 Rows can be extended to negative k with T(n, -k) = T(n, k). Sums of such extended rows give A002898.
%e A353091 The triangle begins:
%e A353091 [1]
%e A353091 []
%e A353091 [6]
%e A353091 [0, 6]
%e A353091 [66, 0, 12]
%e A353091 [0, 150, 0, 30]
%e A353091 [1020, 0, 420, 0, 84, 0, 6]
%e A353091 [0, 3444, 0, 1302, 0, 252, 0, 42]
%e A353091 [19890, 0, 11952, 0, 4284, 0, 984, 0, 216, 0, 24]
%e A353091 [0, 82062, 0, 42972, 0, 14814, 0, 4248, 0, 990, 0, 216, 0, 18]
%e A353091 [449976, 0, 327420, 0, 158970, 0, 57180, 0, 18780, 0, 5190, 0, 1350, 0, 270, 0, 30]
%e A353091 ...
%Y A353091 Cf. A069813 (greatest area), A002898 (all closed walks), A352838 (square lattice).
%Y A353091 For n > 1, row n seems to end with A109047(n).
%K A353091 nonn,tabf,walk
%O A353091 0,2
%A A353091 _Andrey Zabolotskiy_, Apr 22 2022