A185287 R(m,n) is the number of ways to split two strings x and y of length m and n, respectively, into the same number of nonempty parts such that at least one of the corresponding parts has length 1 and such that the parts of the y string have at most size 2.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 3, 0, 0, 1, 4, 5, 3, 0, 0, 1, 5, 8, 7, 3, 0, 0, 1, 6, 12, 13, 7, 0, 0, 0, 1, 7, 17, 22, 16, 6, 0, 0, 0, 1, 8, 23, 35, 32, 17, 4, 0, 0, 0, 1, 9, 30, 53, 58, 39, 14, 0, 0, 0, 0, 1, 10, 38, 77, 98, 80, 40, 10, 0, 0

Offset: 1

Author

Steffen Eger, Feb 20 2011

Keywords

• nonn
• tabl
• nice

Examples

			1    0    0    0    0    0    0    0    0    0    0    0
1    1    2    0    0    0    0    0    0    0    0    0
1    2    3    3    3    0    0    0    0    0    0    0
1    3    5    7    7    6    4    0    0    0    0    0
1    4    8   13   16   17   14   10    5    0    0    0
1    5   12   22   32   39   40   35   25   15    6    0
1    6   17   35   58   80   95   97   86   65   41   21
1    7   23   53   98  151  201  233  238  213  167  112
1    8   30   77  157  267  392  505  577  587  532  427
1    9   38  108  241  448  718 1013 1273 1436 1458 1333
1   10   47  147  357  720 1250 1912 2612 3217 3590 3640
1   11   57  195  513 1116 2086 3434 5056 6728 8146 9011
		

Crossrefs

Cf. A180091.

Programs

Mathematica

r[m_, n_] := Binomial[m-1, n-1] + Sum[ Binomial[k, 2k-n]*Binomial[k+m-n-1, 2k-n-1], {k, 2, n-1}]; r[m_, n_] /; n > 2m-1 = 0; Flatten[ Table[ r[m-k+1, k], {m, 1, 12}, {k, 1, m}]] (* Jean-François Alcover, Nov 07 2011 *)

PARI

C(n,k)=if(nJoerg Arndt, Mar 11 2011 */

Formula

R(m,n) = C(m-1,n-1) + Sum_{k=2..n-1} C(m+k-n-1,2*k-n-1)*C(k,2*k-n).