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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.

%I A349147 #16 Jun 10 2024 08:43:25
%S A349147 1,1,4,1,7,18,1,10,34,80,1,13,55,155,350,1,16,81,266,686,1512,1,19,
%T A349147 112,420,1218,2982,6468,1,22,148,624,2010,5412,12804,27456,1,25,189,
%U A349147 885,3135,9207,23595,54483,115830,1,28,235,1210,4675,14872,41041,101530,230230,486200,1,31
%N 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.
%H A349147 R. J. Mathar, <a href="/A349147/a349147.pdf">The Number of Runs of Words on a 2-letter Alphabet</a> (2021)
%F A349147 T(n,m) = T(m,n).
%F A349147 Sum_{m=0..n} T(n,m) = A000917(n-1) + A000984(n) = 1, 5, 26, 125, 574, ... - _R. J. Mathar_, Nov 09 2021
%F A349147 T(n,m) = binomial(n+m,n)*(2*n*m+n+m)/(n+m) for n+m >= 1.
%e A349147 The triangle starts
%e A349147   1,
%e A349147   1,  4,
%e A349147   1,  7,  18,
%e A349147   1, 10,  34,   80,
%e A349147   1, 13,  55,  155,  350,
%e A349147   1, 16,  81,  266,  686,  1512,
%e A349147   1, 19, 112,  420, 1218,  2982,  6468,
%e A349147   1, 22, 148,  624, 2010,  5412, 12804,  27456,
%e A349147   1, 25, 189,  885, 3135,  9207, 23595,  54483, 115830,
%e A349147   1, 28, 235, 1210, 4675, 14872, 41041, 101530, 230230, 486200,
%e A349147   1, 31, 286, 1606, 6721, 23023, 68068, 179608, 432718, 967538, 2032316
%e A349147 For n=m=1 the sequences are ab (2 runs) and ba (2 runs), so T(1,1)=2+2=4.
%e A349147 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.
%e A349147 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.
%Y A349147 Cf. A016777 (row/col 1), A000566 (row/col 2), A007584 (row/col 3), A051798 (row/col 4).
%Y A349147 Diagonal gives A037965(n+1).
%K A349147 nonn,tabl,easy
%O A349147 0,3
%A A349147 _R. J. Mathar_, Nov 08 2021