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Triangle read by rows: T(n,m) = number of solutions x_1 + x_2 + ... + x_k <= n, where 1 <= x_i <= m, and any k >= 1.
1

%I #13 Nov 04 2019 02:21:43

%S 1,2,3,3,6,7,4,11,14,15,5,19,27,30,31,6,32,51,59,62,63,7,53,95,115,

%T 123,126,127,8,87,176,223,243,251,254,255,9,142,325,431,479,499,507,

%U 510,511,10,231,599,832,943,991,1011,1019,1022,1023,11,375,1103,1605

%N Triangle read by rows: T(n,m) = number of solutions x_1 + x_2 + ... + x_k <= n, where 1 <= x_i <= m, and any k >= 1.

%F For a fixed m, generating function is 1/(1-2*x+x^(m+1)) - 1/(1-x).

%F T(n,m) = Sum_{i=0..floor(n/(m+1))} binomial(n-mi, i)*(-1)^i*2^(n-(m+1)i) - 1.

%F T(n,m) = 2^m - 1 + Sum_{j=m+1..n} A126198(j,m).

%e Triangle begins:

%e 1;

%e 2, 3;

%e 3, 6, 7;

%e 4, 11, 14, 15;

%e 5, 19, 27, 30, 31;

%e 6, 32, 51, 59, 62, 63;

%e 7, 53, 95, 115, 123, 126, 127;

%e ...

%e Could also be extended to a square array:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 2, 3, 3, 3, 3, 3, 3, ...

%e 3, 6, 7, 7, 7, 7, 7, ...

%e 4, 11, 14, 15, 15, 15, 15, ...

%e 5, 19, 27, 30, 31, 31, 31, ...

%e 6, 32, 51, 59, 62, 63, 63, ...

%e 7, 53, 95, 115, 123, 126, 127, ...

%o (PARI) { T(n,m) = sum(i=0, n\(m+1), binomial(n-m*i,i) * (-1)^i * 2^(n-(m+1)*i) ) - 1 }

%Y Cf. A001911 (second column), A027084 (third column), A126198.

%K nonn,tabl

%O 1,2

%A _Max Alekseyev_, Nov 17 2010