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A339011
Sum over all partitions of n of the product of the number of parts and the number of distinct parts.
4
0, 1, 3, 8, 17, 34, 61, 107, 176, 284, 442, 676, 1007, 1483, 2140, 3055, 4299, 5993, 8255, 11284, 15272, 20529, 27373, 36274, 47735, 62484, 81293, 105251, 135555, 173818, 221836, 282003, 356980, 450256, 565765, 708537, 884296, 1100287, 1364736, 1687952, 2081724
OFFSET
0,3
LINKS
MAPLE
b:= proc(n, i, p, d) option remember; `if`(n=0, d*p, `if`(i<1, 0,
add(b(n-i*j, i-1, p+j, `if`(j=0, d, d+1)), j=0..n/i)))
end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n<=0 or i=0, [0$2],
`if`(i=1, [1, n], b(n, i-1)+ (p-> p+[0, p[1]])(b(n-i, i))))
end:
a:= proc(n) option remember; b(n$2)[2]+`if`(n<0, 0, a(n-1)) end:
seq(a(n), n=0..100); # Alois P. Heinz, Jul 25 2022
MATHEMATICA
b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, d*p, If[i < 1, 0,
Sum[b[n - i*j, i - 1, p + j, If[j == 0, d, d + 1]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 09 2021, after Alois P. Heinz *)
CROSSREFS
Essentially partial sums of A093694.
Sequence in context: A281166 A165273 A002626 * A029859 A344004 A305105
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Nov 18 2020
STATUS
approved