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