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 A161224 Triangular table a(n,m) that counts the occurrences of m in all partitions of 2n in exactly n parts. 2
 0, 0, 1, 1, 2, 1, 3, 4, 1, 1, 8, 7, 3, 1, 1, 15, 12, 4, 2, 1, 1, 31, 19, 8, 4, 2, 1, 1, 51, 30, 11, 6, 3, 2, 1, 1, 90, 45, 19, 9, 6, 3, 2, 1, 1, 142, 67, 26, 15, 8, 5, 3, 2, 1, 1, 228, 97, 41, 21, 13, 8, 5, 3, 2, 1, 1, 341, 139, 56, 31, 18, 12, 7, 5, 3, 2, 1, 1, 525, 195, 83, 45, 28, 17, 12, 7, 5, 3, 2, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums are A066186, or n*p(n) with p(n) = A000041 = the partitions of n. The rows reversed converge to 1,1,2,3,5,7,11,15,... or p(n). The count of partitions of 2n in exactly n parts equals p(n). It appears the row n lists A196087(n) together with the row n of triangle A066633. - Omar E. Pol, Feb 26 2012 LINKS Alois P. Heinz, Rows n = 0..140, flattened Eric Weisstein's World of Mathematics, Elder's Theorem EXAMPLE Table starts: 0; 0, 1; 1, 2, 1; 3, 4, 1, 1; 8, 7, 3, 1, 1; since the strict partitions of (2 in 1 part) is {2} with 0 "1" and 1 "2" (4 in 2 parts) is {2,2}, {3,1} with1 "1", 2 "2" and 1 "3" (6 in 3 parts) is {2,2,2}, {3,2,1}, {4,1,1} with 3 "1", 4 "2", 1 "3" and 1 "4" MAPLE b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),       `if`(i=1, `if`(t=n, 1+t*x, 0), expand(add((p->p+coeff(       p, x, 0)*j*x^i)(b(n-i*j, i-1, t-j)), j=0..min(t, n/i)))))     end: a:= n->(p->seq(coeff(p, x, i), i=1..n+1))(b(2*n\$2, n)): seq(a(n), n=0..12);  # Alois P. Heinz, Feb 11 2014 MATHEMATICA <

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Last modified May 20 03:10 EDT 2019. Contains 323412 sequences. (Running on oeis4.)