A339011 - OEIS

%I #21 Jul 25 2022 09:25:33

%S 0,1,3,8,17,34,61,107,176,284,442,676,1007,1483,2140,3055,4299,5993,

%T 8255,11284,15272,20529,27373,36274,47735,62484,81293,105251,135555,

%U 173818,221836,282003,356980,450256,565765,708537,884296,1100287,1364736,1687952,2081724

%N Sum over all partitions of n of the product of the number of parts and the number of distinct parts.

%H Alois P. Heinz, <a href="/A339011/b339011.txt">Table of n, a(n) for n = 0..5000</a>

%p b:= proc(n, i, p, d) option remember; `if`(n=0, d*p, `if`(i<1, 0,

%p       add(b(n-i*j, i-1, p+j, `if`(j=0, d, d+1)), j=0..n/i)))

%p     end:

%p a:= n-> b(n$2, 0$2):

%p seq(a(n), n=0..50);

%p # second Maple program:

%p b:= proc(n, i) option remember; `if`(n<=0 or i=0, [0$2],

%p      `if`(i=1, [1, n], b(n, i-1)+ (p-> p+[0, p[1]])(b(n-i, i))))

%p     end:

%p a:= proc(n) option remember; b(n$2)[2]+`if`(n<0, 0, a(n-1)) end:

%p seq(a(n), n=0..100);  # _Alois P. Heinz_, Jul 25 2022

%t b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, d*p, If[i < 1, 0,

%t      Sum[b[n - i*j, i - 1, p + j, If[j == 0, d, d + 1]], {j, 0, n/i}]]];

%t a[n_] := b[n, n, 0, 0];

%t a /@ Range[0, 50] (* _Jean-François Alcover_, Mar 09 2021, after _Alois P. Heinz_ *)

%Y Cf. A096541, A339006, A339312.

%Y Essentially partial sums of A093694.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Nov 18 2020