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 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 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. 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. EXAMPLE 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;   ... PROG (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); CROSSREFS Main diagonal: A000108. Cf. A007318, A037306, A046854, A047996, A065941, A241926, A267632. Sequence in context: A208245 A309049 A274190 * A037306 A194799 A291119 Adjacent sequences:  A322593 A322594 A322595 * A322597 A322598 A322599 KEYWORD nonn,easy,tabl AUTHOR Franck Maminirina Ramaharo, Jan 22 2019 STATUS approved

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Last modified July 30 07:58 EDT 2021. Contains 346348 sequences. (Running on oeis4.)