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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.

%I A185287 #32 Dec 12 2015 13:11:47
%S A185287 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,
%T A185287 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,
%U A185287 14,0,0,0,0,1,10,38,77,98,80,40,10,0,0
%N 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.
%F A185287 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).
%e A185287 1    0    0    0    0    0    0    0    0    0    0    0
%e A185287 1    1    2    0    0    0    0    0    0    0    0    0
%e A185287 1    2    3    3    3    0    0    0    0    0    0    0
%e A185287 1    3    5    7    7    6    4    0    0    0    0    0
%e A185287 1    4    8   13   16   17   14   10    5    0    0    0
%e A185287 1    5   12   22   32   39   40   35   25   15    6    0
%e A185287 1    6   17   35   58   80   95   97   86   65   41   21
%e A185287 1    7   23   53   98  151  201  233  238  213  167  112
%e A185287 1    8   30   77  157  267  392  505  577  587  532  427
%e A185287 1    9   38  108  241  448  718 1013 1273 1436 1458 1333
%e A185287 1   10   47  147  357  720 1250 1912 2612 3217 3590 3640
%e A185287 1   11   57  195  513 1116 2086 3434 5056 6728 8146 9011
%t A185287 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 *)
%o A185287 (PARI)
%o A185287 C(n,k)=if(n<k, 0, binomial(n,k));
%o A185287 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) );
%o A185287 for (d=1,14, for(c=1,d,print1(R(d-c+1,c),",")))
%o A185287 /* _Joerg Arndt_, Mar 11 2011 */
%Y A185287 Cf. A180091.
%K A185287 nonn,tabl,nice
%O A185287 1,8
%A A185287 _Steffen Eger_, Feb 20 2011