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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.
1
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
1,8
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).
EXAMPLE
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
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 *)
PROG
(PARI)
C(n, k)=if(n<k, 0, binomial(n, k));
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) );
for (d=1, 14, for(c=1, d, print1(R(d-c+1, c), ", ")))
/* Joerg Arndt, Mar 11 2011 */
CROSSREFS
Cf. A180091.
Sequence in context: A296339 A362686 A104245 * A276554 A297323 A257654
KEYWORD
nonn,tabl,nice
AUTHOR
Steffen Eger, Feb 20 2011
STATUS
approved