A349147 Triangle T(n,m) read by rows: the sum of runs of all sequences arranging n objects of one type and m objects of another type.

1, 1, 4, 1, 7, 18, 1, 10, 34, 80, 1, 13, 55, 155, 350, 1, 16, 81, 266, 686, 1512, 1, 19, 112, 420, 1218, 2982, 6468, 1, 22, 148, 624, 2010, 5412, 12804, 27456, 1, 25, 189, 885, 3135, 9207, 23595, 54483, 115830, 1, 28, 235, 1210, 4675, 14872, 41041, 101530, 230230, 486200, 1, 31

Offset

0,3

Links

Table of n, a(n) for n=0..56.

R. J. Mathar, The Number of Runs of Words on a 2-letter Alphabet (2021)

Formula

T(n,m) = T(m,n).

Sum_{m=0..n} T(n,m) = A000917(n-1) + A000984(n) = 1, 5, 26, 125, 574,... - R. J. Mathar, Nov 09 2021

T(n,m) = binomial(n+m,n)*(2*n*m+n+m)/(n+m) for n+m>=1.

Example

The triangle starts
1,
1,  4,
1,  7,  18,
1, 10,  34,   80,
1, 13,  55,  155,  350,
1, 16,  81,  266,  686,  1512,
1, 19, 112,  420, 1218,  2982,  6468,
1, 22, 148,  624, 2010,  5412, 12804,  27456,
1, 25, 189,  885, 3135,  9207, 23595,  54483, 115830,
1, 28, 235, 1210, 4675, 14872, 41041, 101530, 230230, 486200,
1, 31, 286, 1606, 6721, 23023, 68068, 179608, 432718, 967538, 2032316
For n=m=1 the sequences are ab (2runs) and ba (2 runs), so T(1,1)=2+2=4.
For n=1, m=2 the sequences are aab (2 runs), aba (3 runs), baa (2 runs), so T(1,2)=2+3+2=7.
For n=m=2 the sequences are aabb (2 runs), abab (4 runs), abba (3 runs), baab (3 runs), baba (4 runs), bbaa (2 runs), so T(2,2) = 2+4+3+3+4+2=18.

Crossrefs

Cf. A016777 (row/col 1), A000566 (row/col 2), A007584 (row/col 3), A051798 (row/col 4).

Diagonal gives A037965(n+1).

Sequence in context: A193842 A134250 A139045 * A262361 A294791 A084884

Adjacent sequences: A349144 A349145 A349146 * A349148 A349149 A349150

Keyword

nonn,tabl,easy

Author

R. J. Mathar, Nov 08 2021

Status

approved