A181695 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, 2, 3, 3, 6, 7, 4, 11, 14, 15, 5, 19, 27, 30, 31, 6, 32, 51, 59, 62, 63, 7, 53, 95, 115, 123, 126, 127, 8, 87, 176, 223, 243, 251, 254, 255, 9, 142, 325, 431, 479, 499, 507, 510, 511, 10, 231, 599, 832, 943, 991, 1011, 1019, 1022, 1023, 11, 375, 1103, 1605

Offset: 1

Author

Max Alekseyev, Nov 17 2010

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  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, 123, 126, 127;
  ...
Could also be extended to a square array:
  1,   1,   1,   1,   1,   1,   1, ...
  2,   3,   3,   3,   3,   3,   3, ...
  3,   6,   7,   7,   7,   7,   7, ...
  4,  11,  14,  15,  15,  15,  15, ...
  5,  19,  27,  30,  31,  31,  31, ...
  6,  32,  51,  59,  62,  63,  63, ...
  7,  53,  95, 115, 123, 126, 127, ...
		

Crossrefs

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

Programs

PARI

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

Formula

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

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

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