A322428 - OEIS

%I #22 Aug 28 2020 16:46:00

%S 1,3,1,8,3,1,19,8,4,1,43,20,11,5,1,94,48,27,16,6,1,202,110,64,42,22,7,

%T 1,428,245,149,100,64,29,8,1,899,533,341,228,163,93,37,9,1,1875,1142,

%U 765,512,383,256,130,46,10,1,3890,2420,1683,1144,859,638,386,176,56,11,1

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

%H Alois P. Heinz, <a href="/A322428/b322428.txt">Rows n = 1..50, flattened</a>

%e The 4 compositions of 3 are: 111, 12, 21, 3.  The sums of k-th largest parts for k=1..3 give: 1+2+2+3 = 8, 1+1+1+0 = 3, 1+0+0+0 = 1.

%e Triangle T(n,k) begins:

%e      1;

%e      3,    1;

%e      8,    3,   1;

%e     19,    8,   4,   1;

%e     43,   20,  11,   5,   1;

%e     94,   48,  27,  16,   6,   1;

%e    202,  110,  64,  42,  22,   7,   1;

%e    428,  245, 149, 100,  64,  29,   8,  1;

%e    899,  533, 341, 228, 163,  93,  37,  9,  1;

%e   1875, 1142, 765, 512, 383, 256, 130, 46, 10, 1;

%e   ...

%p b:= proc(n, l) option remember; `if`(n=0, add(l[-i]*x^i,

%p       i=1..nops(l)), add(b(n-j, sort([l[], j])), j=1..n))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, [])):

%p seq(T(n), n=1..12);

%t 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 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][ b[n, {}]];

%t Array[T, 12] // Flatten (* _Jean-François Alcover_, Dec 29 2018, after _Alois P. Heinz_ *)

%Y Column k=1 gives A102712.

%Y Row sums give A001787.

%Y T(n+1,1+ceiling(n/2)) gives A027306.

%Y Cf. A322427.

%K nonn,tabl

%O 1,2

%A _Alois P. Heinz_, Dec 07 2018