A252866 Number T(n,k) of parts p in all partitions of n with largest integer power k (such that A052409(p)=k); triangle T(n,k), n>=1, 0<=k<=A000523(n), read by rows.

1, 2, 1, 4, 2, 7, 4, 1, 12, 7, 1, 19, 14, 2, 30, 21, 3, 45, 34, 6, 1, 67, 51, 9, 1, 97, 79, 14, 2, 139, 113, 20, 3, 195, 168, 31, 5, 272, 234, 43, 7, 373, 334, 62, 11, 508, 460, 85, 15, 684, 635, 120, 23, 1, 915, 857, 161, 31, 1, 1212, 1165, 221, 44, 2, 1597

Offset: 1

Author

Alois P. Heinz, Dec 23 2014

Keywords

• nonn
• tabf
• look

Examples

			Triangle T(n,k) begins:
01:    1;
02:    2,   1;
03:    4,   2;
04:    7,   4,   1;
05:   12,   7,   1;
06:   19,  14,   2;
07:   30,  21,   3;
08:   45,  34,   6,  1;
09:   67,  51,   9,  1;
10:   97,  79,  14,  2;
11:  139, 113,  20,  3;
12:  195, 168,  31,  5;
13:  272, 234,  43,  7;
14:  373, 334,  62, 11;
15:  508, 460,  85, 15;
16:  684, 635, 120, 23,  1;
		

Links

• Alois P. Heinz, Rows n = 1..2048, flattened

Crossrefs

Column k=0 gives A000070(n-1).

Row sums give: A006128.

Cf. A000523, A052409.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((p-> p+[0, p[1]*j*x^igcd(seq(h[2], h=ifactors(i)[2]))]
      )(b(n-i*j, i-1)), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)[2]):
seq(T(n), n=1..25);

Formula

T(2^k,k) = 1.