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 A063995 Triangle read by rows: T(n,k), n >= 1, -(n-1) <= k <= n-1, = number of partitions of n with rank k. 12
 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 3, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 2, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 4, 3, 5, 4, 5, 3, 4, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,30 COMMENTS The rank of a partition is the largest part minus the number of parts. The rows are symmetric: for every partition of rank r there is its conjugate with rank -r. [Joerg Arndt, Oct 07 2012] REFERENCES Atkin, A. O. L. and Swinnerton-Dyer, P., Some properties of partitions, Proc. London Math. Soc. (3) 4, (1954). 84-106. Math. Rev. 15,685d. LINKS EXAMPLE The partition 5 = 4+1 has largest summand 4 and 2 summands, hence has rank 4-2 = 2. Triangle begins: [ 1]                               1, [ 2]                            1, 0, 1, [ 3]                         1, 0, 1, 0, 1, [ 4]                      1, 0, 1, 1, 1, 0, 1, [ 5]                   1, 0, 1, 1, 1, 1, 1, 0, 1, [ 6]                1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, [ 7]             1, 0, 1, 1, 2, 1, 3, 1, 2, 1, 1, 0, 1, [ 8]          1, 0, 1, 1, 2, 2, 3, 2, 3, 2, 2, 1, 1, 0, 1, [ 9]       1, 0, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 0, 1, [10]    1, 0, 1, 1, 2, 2, 4, 3, 5, 4, 5, 3, 4, 2, 2, 1, 1, 0, 1, [11] 1, 0, 1, 1, 2, ... Row 20 is: T(20, k) = 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 20, 22, 30, 33, 40, 42, 48, 45, 48, 42, 40, 33, 30, 22, 20, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1; -19 <= k <= 19. MATHEMATICA Table[ Count[ (First[ # ]-Length[ # ]& /@ Partitions[ k ]), # ]&/(AT)Range[ -k+1, k-1 ], {k, 16} ] CROSSREFS For the number of partitions of n with rank 0 (balanced partitions) see A047993. Sequence in context: A052308 A116510 A128915 * A020951 A117118 A117168 Adjacent sequences:  A063992 A063993 A063994 * A063996 A063997 A063998 KEYWORD nonn,nice,tabf,changed AUTHOR N. J. A. Sloane, Sep 19 2001 EXTENSIONS More terms from Vladeta Jovovic and Wouter Meeussen, Sep 19 2001 STATUS approved

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