A107702 Triangle related to guillotine partitions of a k-dimensional box by n hyperplanes.

1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 15, 22, 1, 1, 5, 28, 93, 90, 1, 1, 6, 45, 244, 645, 394, 1, 1, 7, 66, 505, 2380, 4791, 1806, 1, 1, 8, 91, 906, 6345, 24868, 37275, 8558, 1, 1, 9, 120, 1477, 13926, 85405, 272188, 299865, 41586, 1, 1, 10, 153, 2248, 26845, 229326, 1204245, 3080596, 2474025, 206098, 1

Offset: 0

Author

Paul Barry, May 21 2005

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A107703. Transpose of square array A103209, read by antidiagonals.

Examples

			Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  6,   1;
  1, 4, 15,  22,    1;
  1, 5, 28,  93,   90,     1;
  1, 6, 45, 244,  645,   394,     1;
  1, 7, 66, 505, 2380,  4791,  1806,    1;
  1, 8, 91, 906, 6345, 24868, 37275, 8558, 1;
  ...
		

Links

• Seiichi Manyama, Rows n = 0..139, flattened
• E. Ackerman, G. Barequet, R. Y. Pinter and D. Romik, The number of guillotine partitions in d dimensions, Inf. Proc. Lett 98 (4) (2006) 162-167.

Crossrefs

Diagonals: A000012, A006318, A103210, A103211, A133305, A133306, A133307, A133308, A133309. - Philippe Deléham, Dec 10 2008

Cf. A000384, A103209.

Programs

PARI

T(n, k) = sum(j=0, k, (n-k)^j*binomial(k+j, 2*j)*binomial(2*j, j)/(j+1)); \\ Seiichi Manyama, Oct 02 2023

Formula

Number triangle T(n, k)=if(k<=n, sum{j=0..k, C(k+j, 2j)(n-k)^j*C(j)}, 0), C(n) given by A000108.