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 A181187 Triangle read by rows: T(n,k) = sum of k-th largest elements in all partitions of n. 42
 1, 3, 1, 6, 2, 1, 12, 5, 2, 1, 20, 8, 4, 2, 1, 35, 16, 8, 4, 2, 1, 54, 24, 13, 7, 4, 2, 1, 86, 41, 22, 13, 7, 4, 2, 1, 128, 61, 35, 20, 12, 7, 4, 2, 1, 192, 95, 54, 33, 20, 12, 7, 4, 2, 1, 275, 136, 80, 49, 31, 19, 12, 7, 4, 2, 1, 399, 204, 121, 76, 48, 31, 19, 12, 7, 4, 2, 1, 556, 284 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS For the connection with A066897 and A066898 see A206563. - Omar E. Pol, Feb 13 2012 T(n,k) is also the total number of parts >= k in all partitions of n. - Omar E. Pol, Feb 14 2012 The first differences of row n together with 1 give the row n of triangle A066633. - Omar E. Pol, Feb 26 2012 We define the k-th rank of a partition as the k-th part minus the number of parts >= k. Since the first part of a partition is also the largest part of the same partition so the Dyson's rank of a partition is the case for k = 1. It appears that the sum of the k-th ranks of all partitions of n is equal to zero. - Omar E. Pol, Mar 04 2012 LINKS Alois P. Heinz, Rows n = 1..141, flattened FORMULA T(n,k) = sum{j=1..n} A207031(j,k). - Omar E. Pol, May 02 2012 EXAMPLE From Omar E. Pol, Feb 13 2012: (Start) Illustration of initial terms. First five rows of triangle as sums of columns from the partitions of the first five positive integers: . .                            5 .                            3+2 .                  4         4+1 .                  2+2       2+2+1 .          3       3+1       3+1+1 .     2    2+1     2+1+1     2+1+1+1 .  1  1+1  1+1+1   1+1+1+1   1+1+1+1+1 . ------------------------------------- .  1, 3,1, 6,2,1, 12,5,2,1, 20,8,4,2,1 --> This triangle .  |  |/|  |/|/|   |/|/|/|   |/|/|/|/| .  1, 2,1, 4,1,1,  7,3,1,1, 12,4,2,1,1 --> A066633 . For more information see A207031 and A206563. ... Triangle begins:     1;     3,   1;     6,   2,   1;    12,   5,   2,  1;    20,   8,   4,  2,  1;    35,  16,   8,  4,  2,  1;    54,  24,  13,  7,  4,  2,  1;    86,  41,  22, 13,  7,  4,  2,  1;   128,  61,  35, 20, 12,  7,  4,  2, 1;   192,  95,  54, 33, 20, 12,  7,  4, 2, 1;   275, 136,  80, 49, 31, 19, 12,  7, 4, 2, 1;   399, 204, 121, 76, 48, 31, 19, 12, 7, 4, 2, 1; (End) MAPLE p:= (f, g)-> zip((x, y)-> x+y, f, g, 0): b:= proc(n, i) option remember; local f, g;       if n=0 or i=1 then [1, n]     else f:= b(n, i-1); g:= `if`(i>n, [0], b(n-i, i));          p(p(f, g), [0\$i, g[1]])       fi     end: T:= proc(n) local j, l, r, t;       l, r, t:= b(n, n), 1, 1;       for j from n to 2 by -1 do t:= t+l[j]; r:=r, t od;       seq([r][1+n-j], j=1..n)     end: seq(T(n), n=1..14); # Alois P. Heinz, Apr 05 2012 MATHEMATICA Table[Plus @@ (PadRight[ #, n]& /@ IntegerPartitions[n]), {n, 16}] T[n_, n_] = 1; T[n_, k_] /; k

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Last modified April 20 16:17 EDT 2019. Contains 322310 sequences. (Running on oeis4.)