A079308 For a partition P of a positive integer, let f(P) be the product of k+1, over all parts k in P. Let a(n,r) be the sum of f(P) over all partitions P of n with smallest part r. Sequence gives table of a(n,r) for 1 <= r <= n, in the order a(1,1); a(2,1), a(2,2); a(3,1), a(3,2), a(3,3); ...

2, 4, 3, 14, 0, 4, 36, 9, 0, 5, 100, 12, 0, 0, 6, 236, 42, 16, 0, 0, 7, 602, 54, 20, 0, 0, 0, 8, 1368, 195, 24, 25, 0, 0, 0, 9, 3242, 246, 92, 30, 0, 0, 0, 0, 10, 7240, 759, 112, 35, 36, 0, 0, 0, 0, 11, 16386, 1134, 232, 40, 42, 0, 0, 0, 0, 0, 12, 35692, 2859, 528, 170, 48, 49, 0, 0, 0, 0, 0, 13

Offset: 1

Author

Alford Arnold, Feb 09 2003

Keywords

• easy
• nonn
• tabl

Examples

			The partitions with minimal part 3 begin 3, 3+3, 4+3, 5+3, 6+3, 3+3+3, ... which yield the following values of f: 4, 16, 20, 24, 28, 64, ... therefore the 3rd column of our table begins 4,0,0,16,20,24,(28+64)=92,...
Triangle a(n,r) begins:
:    2;
:    4,   3;
:   14,   0,   4;
:   36,   9,   0,  5;
:  100,  12,   0,  0,  6;
:  236,  42,  16,  0,  0, 7;
:  602,  54,  20,  0,  0, 0, 8;
: 1368, 195,  24, 25,  0, 0, 0, 9;
: 3242, 246,  92, 30,  0, 0, 0, 0, 10;
: 7240, 759, 112, 35, 36, 0, 0, 0,  0, 11;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Cf. A074139, A074141 (row sums).

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1,
      `if`(k>n, 0, b(n, k+1) +(k+1)*b(n-k, k)))
    end:
a:= (n, k)-> b(n,k)-b(n, k+1):
seq(seq(a(n, k), k=1..n), n=1..12);  # Alois P. Heinz, May 22 2015

Mathematica

a[n_, r_] := Which[r>n, 0, r==n, n+1, True, a[n, r]=(r+1)Sum[a[n-r, s], {s, r, n-r}]]; Flatten[Table[a[n, r], {n, 1, 12}, {r, 1, n}]]

Extensions

Edited by Dean Hickerson, Feb 11 2003

Offset changed to 1 by Alois P. Heinz, May 22 2015