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%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
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