The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A321721 Number of non-isomorphic non-normal semi-magic square multiset partitions of weight n. 14
 1, 1, 2, 2, 4, 2, 7, 2, 10, 7, 12, 2, 38, 2, 21, 46, 72, 2, 162, 2, 420, 415, 64, 2, 4987, 1858, 110, 9336, 45456, 2, 136018, 2, 1014658, 406578, 308, 3996977, 34937078, 2, 502, 28010167, 1530292965, 2, 508164038, 2, 54902992348, 51712929897, 1269, 2, 3217847072904, 8597641914, 9168720349613 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A non-normal semi-magic square multiset partition of weight n is a multiset partition of weight n whose part sizes and vertex degrees are all equal to d, for some d|n. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices. Also the number of nonnegative integer square matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with row sums and column sums all equal to d, for some d|n. LINKS Wikipedia, Magic square FORMULA a(p) = 2 for p prime corresponding to the 1 X 1 square [p] and the permutation matrices of size p X p with partition (1...10...0). - Chai Wah Wu, Jan 16 2019 a(n) = Sum_{d|n} A333733(d,n/d) for n > 0. - Andrew Howroyd, Apr 11 2020 EXAMPLE Non-isomorphic representatives of the a(2) = 2 through a(6) = 7 multiset partitions:   {{11}}   {{111}}     {{1111}}       {{11111}}         {{111111}}   {{1}{2}} {{1}{2}{3}} {{11}{22}}     {{1}{2}{3}{4}{5}} {{111}{222}}                        {{12}{12}}                       {{112}{122}}                        {{1}{2}{3}{4}}                   {{11}{22}{33}}                                                         {{11}{23}{23}}                                                         {{12}{13}{23}}                                                         {{1}{2}{3}{4}{5}{6}} Inequivalent representatives of the a(6) = 7 matrices:   [6] .   [3 0] [2 1]   [0 3] [1 2] .   [2 0 0] [2 0 0] [1 1 0]   [0 2 0] [0 1 1] [1 0 1]   [0 0 2] [0 1 1] [0 1 1] .   [1 0 0 0 0 0]   [0 1 0 0 0 0]   [0 0 1 0 0 0]   [0 0 0 1 0 0]   [0 0 0 0 1 0]   [0 0 0 0 0 1] Inequivalent representatives of the a(9) = 7 matrices:   [9] .   [3 0 0] [3 0 0] [2 1 0] [2 1 0] [1 1 1]   [0 3 0] [0 2 1] [1 1 1] [1 0 2] [1 1 1]   [0 0 3] [0 1 2] [0 1 2] [0 2 1] [1 1 1] .   [1 0 0 0 0 0 0 0 0]   [0 1 0 0 0 0 0 0 0]   [0 0 1 0 0 0 0 0 0]   [0 0 0 1 0 0 0 0 0]   [0 0 0 0 1 0 0 0 0]   [0 0 0 0 0 1 0 0 0]   [0 0 0 0 0 0 1 0 0]   [0 0 0 0 0 0 0 1 0]   [0 0 0 0 0 0 0 0 1] CROSSREFS Cf. A006052, A007716, A057150, A120732, A271103, A319056, A319616. Cf. A321717, A321718, A321719, A321722, A321724, A333733. Sequence in context: A207329 A122977 A275870 * A003980 A286369 A132801 Adjacent sequences:  A321718 A321719 A321720 * A321722 A321723 A321724 KEYWORD nonn AUTHOR Gus Wiseman, Nov 18 2018 EXTENSIONS a(11)-a(13) from Chai Wah Wu, Jan 16 2019 a(14)-a(15) from Chai Wah Wu, Jan 20 2019 Terms a(16) and beyond from Andrew Howroyd, Apr 11 2020 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 15 00:00 EDT 2021. Contains 345041 sequences. (Running on oeis4.)