A322596 Square array read by descending antidiagonals (n >= 0, k >= 0): let b(n,k) = (n+k)!/((n+1)!*k!); then T(n,k) = b(n,k) if b(n,k) is an integer, and T(n,k) = floor(b(n,k)) + 1 otherwise.

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, 1, 6, 19, 42, 66, 77, 66, 42, 19, 6, 1, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1, 1

Offset: 0

Author

Franck Maminirina Ramaharo, Jan 22 2019

Keywords

• nonn
• easy
• tabl

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.

Examples

			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;
  ...
		

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.

Crossrefs

Main diagonal: A000108.

Cf. A007318, A037306, A046854, A047996, A065941, A241926, A267632.

Programs

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);