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A322428 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. 4
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, 1, 428, 245, 149, 100, 64, 29, 8, 1, 899, 533, 341, 228, 163, 93, 37, 9, 1, 1875, 1142, 765, 512, 383, 256, 130, 46, 10, 1, 3890, 2420, 1683, 1144, 859, 638, 386, 176, 56, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Alois P. Heinz, Rows n = 1..50, flattened

EXAMPLE

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.

Triangle T(n,k) begins:

     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,   1;

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

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

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

MAPLE

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

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

    end:

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

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

MATHEMATICA

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

Array[T, 12] // Flatten (* Jean-Fran├žois Alcover, Dec 29 2018, after Alois P. Heinz *)

CROSSREFS

Column k=1 of A102712.

Row sums give A001787.

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

Cf. A322427.

Sequence in context: A049541 A249757 A207609 * A130300 A308742 A065451

Adjacent sequences:  A322425 A322426 A322427 * A322429 A322430 A322431

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Dec 07 2018

STATUS

approved

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Last modified November 18 09:46 EST 2019. Contains 329261 sequences. (Running on oeis4.)