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A107702
Triangle related to guillotine partitions of a k-dimensional box by n hyperplanes.
2
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
0,5
COMMENTS
Row sums are A107703. Transpose of square array A103209, read by antidiagonals.
LINKS
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.
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.
EXAMPLE
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;
...
PROG
(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
CROSSREFS
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, May 21 2005
STATUS
approved