OFFSET
0,8
COMMENTS
For n >= 1, T(n,k) is the number of nodes in n-dimensional space for Mysovskikh's cubature formula which is exact for any polynomial of degree k of n variables.
LINKS
Ronald Cools, Encyclopaedia of Cubature Formulas
Ivan P. Mysovskikh, On the construction of cubature formulae for very simple domains, USSR Computational Mathematics and Mathematical Physics, Volume 4, Issue 1, 1964, 1-17.
EXAMPLE
Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...
1, 1, 2, 4, 5, 7, 10, 12, 15, 19, ...
1, 1, 3, 5, 9, 14, 21, 30, 42, 55, ...
1, 1, 3, 7, 14, 26, 42, 66, 99, 143, ...
1, 1, 4, 10, 21, 42, 77, 132, 215, 334, ...
1, 1, 4, 12, 30, 66, 132, 246, 429, 715, ...
1, 1, 5, 15, 42, 99, 215, 429, 805, 1430, ...
1, 1, 5, 19, 55, 143, 334, 715, 1430, 2702, ...
1, 1, 6, 22, 72, 201, 501, 1144, 2431, 4862, ...
...
As triangular array, this begins:
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
1, 2, 2, 1, 1;
1, 3, 4, 3, 1, 1;
1, 3, 5, 5, 3, 1, 1;
1, 4, 7, 9, 7, 4, 1, 1;
1, 4, 10, 14, 14, 10, 4, 1, 1;
1, 5, 12, 21, 26, 21, 12, 5, 1, 1;
1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1;
...
PROG
(Maxima)
b(n, k) := (n + k)!/((n + 1)!*k!)$
T(n, k) := if integerp(b(n, k)) then b(n, k) else floor(b(n, k)) + 1$
create_list(T(k, n - k), n, 0, 15, k, 0, n);
CROSSREFS
KEYWORD
AUTHOR
Franck Maminirina Ramaharo, Jan 22 2019
STATUS
approved