A306210 T(n,k) = binomial(n + k, n) - binomial(n + floor(k/2), n) + 1, square array read by descending antidiagonals (n >= 0, k >= 0).

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 4, 4, 1, 1, 3, 8, 7, 5, 1, 1, 4, 10, 17, 11, 6, 1, 1, 4, 16, 26, 31, 16, 7, 1, 1, 5, 19, 47, 56, 51, 22, 8, 1, 1, 5, 27, 65, 112, 106, 78, 29, 9, 1, 1, 6, 31, 101, 176, 232, 183, 113, 37, 10, 1, 1, 6, 41, 131, 296, 407, 435, 295, 157, 46, 11, 1

Offset: 0

Author

Franck Maminirina Ramaharo, Jan 29 2019

Keywords

• nonn
• easy
• tabl

Comments

There are at most T(n,k) possible values for the number of knots in an interpolatory cubature formula of degree k for an integral over an n-dimensional region.

Examples

			Square array begins:
  1, 1,  1,   1,   1,    1,    1,    1,     1,  ...
  1, 2,  2,   3,   3,    4,    4,    5,     5,  ...
  1, 3,  4,   8,  10,   16,   19,   27,    31,  ...
  1, 4,  7,  17,  26,   47,   65,  101,   131,  ...
  1, 5, 11,  31,  56,  112,  176,  296,   426,  ...
  1, 6, 16,  51, 106,  232,  407,  737,  1162,  ...
  1, 7, 22,  78, 183,  435,  841, 1633,  2794,  ...
  1, 8, 29, 113, 295,  757, 1597, 3313,  6106,  ...
  1, 9, 37, 157, 451, 1243, 2839, 6271, 12376,  ...
  ...
As triangular array, this begins:
  1;
  1, 1;
  1, 2,  1;
  1, 2,  3,  1;
  1, 3,  4,  4,  1;
  1, 3,  8,  7,  5,  1;
  1, 4, 10, 17, 11,  6,  1;
  1, 4, 16, 26, 31, 16,  7, 1;
  1, 5, 19, 47, 56, 51, 22, 8, 1;
  ...
		

Links

• Ronald Cools, Encyclopaedia of Cubature Formulas
• Ronald Cools, A Survey of Methods for Constructing Cubature Formulae, In: Espelid T.O., Genz A. (eds), Numerical Integration, NATO ASI Series (Series C: Mathematical and Physical Sciences), Vol. 357, 1991, Springer, Dordrecht, pp. 1-24.
• T. N. L. Patterson, On the Construction of a Practical Ermakov-Zolotukhin Multiple Integrator, In: Keast P., Fairweather G. (eds), Numerical Integration, NATO ASI Series (Series C: Mathematical and Physical Sciences), Vol. 203, 1987, Springer, Dordrecht, pp. 269-290.

Crossrefs

Cf. A007318, A046854, A322596.

Programs

Mathematica

T[n_, k_] = Binomial[n + k, n] - Binomial[n + Floor[k/2], n] + 1;
Table[T[k, n - k], {k, 0, n}, {n, 0, 20}] // Flatten

Maxima

T(n, k) := binomial(n + k, n) - binomial(n + floor(k/2), n) + 1$
create_list(T(k, n - k), n, 0, 20, k, 0, n);

Formula

T(n,k) = A007318(n+k,n) - A046854(n+k,n) + 1.

G.f.: (1 - x - x^2 + x^3 - 2*y + 2*x*y + y^2 - x*y^2 + x^2*y^2)/((1 - x)*(1 - y)*(1 - x - y)*(1 - x^2 - y)).