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%I #10 Feb 15 2021 03:44:44
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%H Ronald Cools, <a href="http://nines.cs.kuleuven.be/ecf/">Encyclopaedia of Cubature Formulas</a>
%H Ivan P. Mysovskikh, <a href="https://doi.org/10.1016/0041-5553(64)90212-5">On the construction of cubature formulae for very simple domains</a>, USSR Computational Mathematics and Mathematical Physics, Volume 4, Issue 1, 1964, 1-17.
%e Array begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...
%e 1, 1, 2, 4, 5, 7, 10, 12, 15, 19, ...
%e 1, 1, 3, 5, 9, 14, 21, 30, 42, 55, ...
%e 1, 1, 3, 7, 14, 26, 42, 66, 99, 143, ...
%e 1, 1, 4, 10, 21, 42, 77, 132, 215, 334, ...
%e 1, 1, 4, 12, 30, 66, 132, 246, 429, 715, ...
%e 1, 1, 5, 15, 42, 99, 215, 429, 805, 1430, ...
%e 1, 1, 5, 19, 55, 143, 334, 715, 1430, 2702, ...
%e 1, 1, 6, 22, 72, 201, 501, 1144, 2431, 4862, ...
%e ...
%e As triangular array, this begins:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 1, 1;
%e 1, 2, 2, 1, 1;
%e 1, 3, 4, 3, 1, 1;
%e 1, 3, 5, 5, 3, 1, 1;
%e 1, 4, 7, 9, 7, 4, 1, 1;
%e 1, 4, 10, 14, 14, 10, 4, 1, 1;
%e 1, 5, 12, 21, 26, 21, 12, 5, 1, 1;
%e 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1;
%e ...
%o (Maxima)
%o b(n, k) := (n + k)!/((n + 1)!*k!)$
%o T(n, k) := if integerp(b(n, k)) then b(n, k) else floor(b(n, k)) + 1$
%o create_list(T(k, n - k), n, 0, 15, k, 0, n);
%Y Main diagonal: A000108.
%Y Cf. A007318, A037306, A046854, A047996, A065941, A241926, A267632.
%K nonn,easy,tabl
%O 0,8
%A _Franck Maminirina Ramaharo_, Jan 22 2019
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