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A336903 Sum of the largest parts of all compositions of n into distinct parts.

%I A336903 #15 Jul 12 2021 04:16:47
%S A336903 0,1,2,7,10,19,42,61,98,151,304,403,654,925,1400,2431,3328,4903,7056,
%T A336903 10117,13952,23419,30406,44683,61308,87289,116822,164359,247774,
%U A336903 327715,457542,624445,855062,1148023,1559188,2058643,3043506,3906637,5375732,7111975,9679852
%N A336903 Sum of the largest parts of all compositions of n into distinct parts.
%H A336903 Alois P. Heinz, <a href="/A336903/b336903.txt">Table of n, a(n) for n = 0..5000</a>
%F A336903 a(n) == n (mod 2).
%e A336903 a(6) = 42 = 3 + 3 + 3 + 3 + 3 + 3 + 4 + 4 + 5 + 5 + 6: 12(3), 1(3)2, 21(3), 2(3)1, (3)12, (3)21, 2(4), (4)2, 1(5), (5)1, (6).
%p A336903 b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
%p A336903       `if`(n=0, p!, b(n-i, min(n-i, i-1), p+1)*
%p A336903       `if`(p=0, i, 1)+b(n, i-1, p)))
%p A336903     end:
%p A336903 a:= n-> `if`(n=0, 0, b(n$2, 0)):
%p A336903 seq(a(n), n=0..50);
%t A336903 b[n_, i_, p_] := b[n, i, p] = If[i(i + 1)/2 < n, 0,
%t A336903      If[n == 0, p!, b[n - i, Min[n - i, i - 1], p + 1]*
%t A336903      If[p == 0, i, 1] + b[n, i - 1, p]]];
%t A336903 a[n_] := If[n == 0, 0, b[n, n, 0]];
%t A336903 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jul 12 2021, after _Alois P. Heinz_ *)
%Y A336903 Cf. A005895, A006128, A046746, A092265, A097939, A102712, A336902.
%K A336903 nonn
%O A336903 0,3
%A A336903 _Alois P. Heinz_, Aug 07 2020