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

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

1,6

Links

Table of n, a(n) for n=1..72.

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

Formula

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.

Example

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,

Prog

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

Crossrefs

Cf. A185356, A202690, A202815, A202816.

Cf. A001586 (row sums).

Sequence in context: A354470 A141692 A261097 * A261217 A352957 A245230

Adjacent sequences: A335332 A335333 A335334 * A335336 A335337 A335338

Keyword

nonn,tabf

Author

Michel Marcus, Jun 02 2020

Status

approved