A202815 -id:A202815 - cp's OEIS frontend

A202816 The left-hand half-triangle of A185356 (or A202690).

1, 1, 0, 3, 4, 4, 11, 8, 4, 0, 57, 68, 76, 80, 80, 361, 304, 236, 160, 80, 0, 2763, 3124, 3428, 3664, 3824, 3904, 3904, 24611, 21848, 18724, 15296, 11632, 7808, 3904, 0, 250737, 275348, 297196, 315920, 331216, 342848, 350656, 354560, 354560

Offset: 1

N. J. A. Sloane, Dec 25 2011

			Triangle begins:
    1
    1   0
    3   4   4
   11   8   4   0
   57  68  76  80 80
  361 304 236 160 80 0
  ...

M. Josuat-Vergès, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272 [math.CO], 2011.

Cf. A185356, A202690, A202815.

T(n,k) = {if ((k==0), return(0)); if (n==1, if (abs(k)==1, return(1))); if (n%2, if (k<0, sum(j=k+1, n-1, T(n-1,j)), sum(j=k, n-1, T(n-1,j))), if (k<0, sum(j=-n+1, k, T(n-1,j)), sum(j=-n+1, k-1, T(n-1,j))));}
tabl(nn) = {for (n=1, nn, for (k=1, n, if (k, print1(T(n, -k), ", "));); print;);} \\ Michel Marcus, Jun 03 2020

More terms from Michel Marcus, Jun 03 2020

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

Original entry on oeis.org

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, 80, 80, 0, 80, 160, 236, 304, 361, 361, 418, 464, 496, 512, 512, 0, 512, 1024, 1520, 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

Offset: 1

Michel Marcus, Jun 02 2020

			Triangle begins:
                        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,  80,  80,
0, 80, 160, 236, 304, 361, 361, 418, 464, 496, 512, 512,

Heesung Shin and Jiang Zeng, More bijections for Entringer and Arnold families, arXiv:2006.00507 [math.CO], 2020.

Cf. A185356, A202690, A202815, A202816.

Cf. A001586 (row sums).

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)));}
tabf(nn) = {for (n=1, nn, for (k=-n, -1, print1(T(n,k), ", ");); for (k=1, n, print1(T(n,k), ", ");); print;);}

T(n,k) is defined by T(1,1) = T(1,-1) = 1, T(n,-n) = 0 (n >= 2), and the recurrence
T(n,k) = T(n,k-1) + T(n-1,-k+1) if n >= k > 1,
T(n,k) = T(n,-1) if n > k = 1,
T(n,k) = T(n,k-1) + T(n-1,-k) if -1 >= k > -n.

cp's OEIS Frontend

A202816 The left-hand half-triangle of A185356 (or A202690).

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