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A317328 Total number of distinct run lengths summed over all permutations of [n] with exactly k distinct lengths of increasing runs.

%I A317328 #13 Apr 15 2022 11:15:41
%S A317328 0,1,2,10,41,238,1560,11822,99884,944951,9851550,112297557,1389182296,
%T A317328 18548151140,265854443194,4071965025912,66379311227650,
%U A317328 1147606112273168,20974771814051529,404111977603746584,8186206878478673819,173948483818885099616,3868878742247603336434,89892706223315981903289
%N A317328 Total number of distinct run lengths summed over all permutations of [n] with exactly k distinct lengths of increasing runs.
%H A317328 Alois P. Heinz, <a href="/A317328/b317328.txt">Table of n, a(n) for n = 0..60</a>
%F A317328 a(n) = Sum_{k=0..floor((sqrt(1+8*n)-1)/2)} k * A317327(n,k).
%p A317328 b:= proc(u, o, t, s) option remember;
%p A317328       `if`(u+o=0, x^(nops(s union {t})-1),
%p A317328        add(b(u-j, o+j-1, 1, s union {t}), j=1..u)+
%p A317328        add(b(u+j-1, o-j, t+1, s), j=1..o))
%p A317328     end:
%p A317328 a:= n-> (p-> add(coeff(p, x, i)*i, i=0..degree(p)))(b(n, 0$2, {})):
%p A317328 seq(a(n), n=0..16);
%t A317328 b[u_, o_, t_, s_] := b[u, o, t, s] =
%t A317328      If[u + o == 0, x^(Length[s ~Union~ {t}] - 1),
%t A317328      Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}], {j, 1, u}] +
%t A317328      Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
%t A317328 a[n_] := Function[p, Sum[Coefficient[p, x, i]*i, {i, 0, Exponent[p, x]}]][
%t A317328      b[n, 0, 0, {}]];
%t A317328 Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Apr 15 2022, after _Alois P. Heinz_ *)
%Y A317328 Cf. A317327.
%K A317328 nonn
%O A317328 0,3
%A A317328 _Alois P. Heinz_, Jul 25 2018