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A322427 Sum T(n,k) of k-th smallest parts of all compositions of n; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%I A322427 #15 Jul 26 2020 11:28:09
%S A322427 1,3,1,6,5,1,12,12,7,1,22,28,20,9,1,42,54,54,30,11,1,79,106,115,92,42,
%T A322427 13,1,151,200,239,218,144,56,15,1,291,376,471,486,378,212,72,17,1,566,
%U A322427 708,904,1014,908,612,298,90,19,1,1106,1346,1709,2030,2014,1584,939,404,110,21,1
%N A322427 Sum T(n,k) of k-th smallest parts of all compositions of n; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%H A322427 Alois P. Heinz, <a href="/A322427/b322427.txt">Rows n = 1..50, flattened</a>
%e A322427 The 4 compositions of 3 are: 111, 12, 21, 3.  The sums of k-th smallest parts for k=1..3 give: 1+1+1+3 = 6, 1+2+2+0 = 5, 1+0+0+0 = 1.
%e A322427 Triangle T(n,k) begins:
%e A322427     1;
%e A322427     3,   1;
%e A322427     6,   5,   1;
%e A322427    12,  12,   7,    1;
%e A322427    22,  28,  20,    9,   1;
%e A322427    42,  54,  54,   30,  11,   1;
%e A322427    79, 106, 115,   92,  42,  13,   1;
%e A322427   151, 200, 239,  218, 144,  56,  15,  1;
%e A322427   291, 376, 471,  486, 378, 212,  72, 17,  1;
%e A322427   566, 708, 904, 1014, 908, 612, 298, 90, 19, 1;
%e A322427   ...
%p A322427 b:= proc(n, l) option remember; `if`(n=0, add(l[i]*x^i,
%p A322427       i=1..nops(l)), add(b(n-j, sort([l[], j])), j=1..n))
%p A322427     end:
%p A322427 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, [])):
%p A322427 seq(T(n), n=1..12);
%t A322427 b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[i]] x^i, {i, 1, Length[l]}], Sum[b[n - j, Sort[Append[l, j]]], {j, 1, n}]];
%t A322427 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][ b[n, {}]];
%t A322427 Array[T, 12] // Flatten (* _Jean-François Alcover_, Dec 29 2018, after _Alois P. Heinz_ *)
%Y A322427 Column k=1 gives A097939.
%Y A322427 Row sums give A001787.
%Y A322427 Cf. A322428.
%K A322427 nonn,tabl
%O A322427 1,2
%A A322427 _Alois P. Heinz_, Dec 07 2018