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A335335 Irregular triangle T(n,k) of Arnold numbers with n>=1 and 1<= abs(k) <= n.

%I A335335 #13 Feb 19 2021 09:51:48
%S A335335 1,1,0,1,1,2,0,2,3,3,4,4,0,4,8,11,11,14,16,16,0,16,32,46,57,57,68,76,
%T A335335 80,80,0,80,160,236,304,361,361,418,464,496,512,512,0,512,1024,1520,
%U A335335 1984,2402,2763,2763,3124,3428,3664,3824,3904,3904,0,3904,7808,11632,15296,18724,21848,24611,24611,27374,29776,31760,33280,34304,34816,34816
%N A335335 Irregular triangle T(n,k) of Arnold numbers with n>=1 and 1<= abs(k) <= n.
%H A335335 Heesung Shin and Jiang Zeng, <a href="https://arxiv.org/abs/2006.00507">More bijections for Entringer and Arnold families</a>, arXiv:2006.00507 [math.CO], 2020.
%F A335335 T(n,k) is defined by T(1,1) = T(1,-1) = 1, T(n,-n) = 0 (n >= 2), and the recurrence
%F A335335 T(n,k) = T(n,k-1) + T(n-1,-k+1) if n >= k > 1,
%F A335335 T(n,k) = T(n,-1) if n > k = 1,
%F A335335 T(n,k) = T(n,k-1) + T(n-1,-k) if -1 >= k > -n.
%e A335335 Triangle begins:
%e A335335                         1,   1,
%e A335335                    0,   1,   1,   2,
%e A335335               0,   2,   3,   3,   4,   4,
%e A335335          0,   4,   8,  11,  11,  14,  16,  16,
%e A335335     0,  16,  32,  46,  57,  57,  68,  76,  80,  80,
%e A335335 0, 80, 160, 236, 304, 361, 361, 418, 464, 496, 512, 512,
%o A335335 (PARI) T(n, k) = {if ((n==1) && (k==1), return (1)); if ((n+k) == 0, if (n==1, return(1), return (0))); if ((n>=k) && (k>1), return(T(n, k-1) + T(n-1, 1-k))); if ((k==1) && (n>k), return(T(n,-1))); if ((-1>=k) && (k>=-n), return(T(n, k-1) + T(n-1, -k)));}
%o A335335 tabf(nn) = {for (n=1, nn, for (k=-n, -1, print1(T(n,k), ", ");); for (k=1, n, print1(T(n,k), ", ");); print;);}
%Y A335335 Cf. A185356, A202690, A202815, A202816.
%Y A335335 Cf. A001586 (row sums).
%K A335335 nonn,tabf
%O A335335 1,6
%A A335335 _Michel Marcus_, Jun 02 2020