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

Author

R. J. Mathar, Nov 08 2021

Keywords

• nonn
• tabl
• easy

Examples

			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 (2 runs) 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.
		

Links

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

Crossrefs

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

Diagonal gives A037965(n+1).

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.