A336875 - OEIS

%I #16 Jun 13 2021 07:16:30

%S 0,1,2,6,13,30,66,144,308,655,1380,2891,6024,12500,25844,53274,109530,

%T 224690,460033,940276,1918979,3911186,7962194,16191875,32896364,

%U 66776727,135445212,274532607,556086916,1125727954,2277650681,4605981879,9310120876,18810538092

%N Number of parts, counted without multiplicity, in all compositions of n.

%H Alois P. Heinz, <a href="/A336875/b336875.txt">Table of n, a(n) for n = 0..1000</a>

%e a(4) = 1 + 2 + 2 + 2 + 1 + 2 + 2 + 1 = 13: (1)111, (1)1(2), (1)(2)1, (2)(1)1, (2)2, (1)(3), (3)(1), (4).

%p b:= proc(n, i, p) option remember; `if`(n=0, [p!, 0],

%p       `if`(i<1, 0, add((p-> [0, `if`(j=0, 0, p[1])]+p)(

%p          b(n-i*j, i-1, p+j)/j!), j=0..n/i)))

%p     end:

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

%p seq(a(n), n=0..38);

%t b[n_, i_, p_] := b[n, i, p] = If[n == 0, {p!, 0},

%t      If[i<1, {0, 0}, Sum[{0, If[j == 0, 0, #[[1]]]}+#&[

%t      b[n-i*j, i-1, p+j]/j!], {j, 0, n/i}]]];

%t a[n_] := b[n, n, 0][[2]];

%t a /@ Range[0, 38] (* _Jean-François Alcover_, Jun 13 2021, after _Alois P. Heinz_ *)

%Y Cf. A000070 (the same for partitions), A001792 (all parts), A097910, A336516.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 06 2020