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%I #7 Feb 04 2019 11:22:43
%S 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,
%T 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,
%U 101,176,232,183,113,37,10,1,1,6,41,131,296,407,435,295,157,46,11,1
%N T(n,k) = binomial(n + k, n) - binomial(n + floor(k/2), n) + 1, square array read by descending antidiagonals (n >= 0, k >= 0).
%C 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.
%H Ronald Cools, <a href="http://nines.cs.kuleuven.be/ecf/">Encyclopaedia of Cubature Formulas</a>
%H Ronald Cools, <a href="https://doi.org/10.1007/978-94-011-2646-5_1">A Survey of Methods for Constructing Cubature Formulae</a>, 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.
%H T. N. L. Patterson, <a href="https://doi.org/10.1007/978-94-009-3889-2_27">On the Construction of a Practical Ermakov-Zolotukhin Multiple Integrator</a>, 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.
%F T(n,k) = A007318(n+k,n) - A046854(n+k,n) + 1.
%F 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)).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 2, 3, 3, 4, 4, 5, 5, ...
%e 1, 3, 4, 8, 10, 16, 19, 27, 31, ...
%e 1, 4, 7, 17, 26, 47, 65, 101, 131, ...
%e 1, 5, 11, 31, 56, 112, 176, 296, 426, ...
%e 1, 6, 16, 51, 106, 232, 407, 737, 1162, ...
%e 1, 7, 22, 78, 183, 435, 841, 1633, 2794, ...
%e 1, 8, 29, 113, 295, 757, 1597, 3313, 6106, ...
%e 1, 9, 37, 157, 451, 1243, 2839, 6271, 12376, ...
%e ...
%e As triangular array, this begins:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 2, 3, 1;
%e 1, 3, 4, 4, 1;
%e 1, 3, 8, 7, 5, 1;
%e 1, 4, 10, 17, 11, 6, 1;
%e 1, 4, 16, 26, 31, 16, 7, 1;
%e 1, 5, 19, 47, 56, 51, 22, 8, 1;
%e ...
%t T[n_, k_] = Binomial[n + k, n] - Binomial[n + Floor[k/2], n] + 1;
%t Table[T[k, n - k], {k, 0, n}, {n, 0, 20}] // Flatten
%o (Maxima)
%o T(n, k) := binomial(n + k, n) - binomial(n + floor(k/2), n) + 1$
%o create_list(T(k, n - k), n, 0, 20, k, 0, n);
%Y Cf. A007318, A046854, A322596.
%K nonn,easy,tabl
%O 0,5
%A _Franck Maminirina Ramaharo_, Jan 29 2019
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