A161224 Triangular table a(n,m) that counts the occurrences of m in all partitions of 2n in exactly n parts.

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

Offset: 0

Author

Wouter Meeussen, Jun 06 2009

Keywords

• nonn
• tabl

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

Examples

			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"
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Eric Weisstein's World of Mathematics, Elder's Theorem

Crossrefs

Cf. A066186, A066633.

Programs

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

<Jean-François Alcover, May 24 2016, after Alois P. Heinz *)

Extensions

Row 0 inserted and tabf changed to tabl by Alois P. Heinz, Feb 11 2014