A123349 Square array of Kekulé numbers for the mirror-symmetrical chevrons Ch(m,n), read by antidiagonals (m,n >= 0).

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 14, 10, 1, 1, 5, 30, 46, 17, 1, 1, 6, 55, 146, 117, 26, 1, 1, 7, 91, 371, 517, 251, 37, 1, 1, 8, 140, 812, 1742, 1476, 478, 50, 1, 1, 9, 204, 1596, 4878, 6376, 3614, 834, 65, 1, 1, 10, 285, 2892, 11934, 22252, 19490, 7890, 1361, 82, 1, 1, 11

Offset

0,5

Comments

T(m,1)=A002522(m); T(m,2)=A123350(m); T(m,3)=A123351(m).

References

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see pp. 119-120).

Links

Table of n, a(n) for n=0..67.

Formula

T(m,n) = Sum_{i=0..n} binomial(m+i-1, i)^2.

Example

T(1,1)=2 because Ch(1,1) consists of a single hexagon; it has 2 perfect matchings: {1,3,5} and {2,4,6}, the edges of the hexagon being labeled consecutively by 1,2,3,4,5,6.
Square array starts:
  1,  1,   1,   1,    1,    1,     1,     1, ...
  1,  2,   3,   4,    5,    6,     7,     8, ...
  1,  5,  14,  30,   55,   91,   140,   204, ...
  1, 10,  46, 146,  371,  812,  1596,  2892, ...
  1, 17, 117, 517, 1742, 4878, 11934, 26334, ...

Maple

T:=(m, n)->sum(binomial(m+i-1, i)^2, i=0..n): TT:=(m, n)->T(m-1, n-1): matrix(9, 9, TT); # yields sequence in matrix form

Crossrefs

Cf. A002522, A123350, A123351.

Sequence in context: A340968 A128198 A320031 * A123352 A114163 A189435

Adjacent sequences: A123346 A123347 A123348 * A123350 A123351 A123352

Keyword

nonn,tabl

Author

N. J. A. Sloane, Oct 14 2006

Extensions

Edited by Emeric Deutsch, Oct 27 2006, Oct 28 2006

Status

approved