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
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.
Programs
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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);
Comments