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A072575 Triangle T(n,k) of number of compositions (ordered partitions) of n into distinct parts where largest part is exactly k, 1<=k<=n. 7
1, 0, 1, 0, 2, 1, 0, 0, 2, 1, 0, 0, 2, 2, 1, 0, 0, 6, 2, 2, 1, 0, 0, 0, 8, 2, 2, 1, 0, 0, 0, 6, 8, 2, 2, 1, 0, 0, 0, 6, 8, 8, 2, 2, 1, 0, 0, 0, 24, 12, 8, 8, 2, 2, 1, 0, 0, 0, 0, 30, 14, 8, 8, 2, 2, 1, 0, 0, 0, 0, 30, 36, 14, 8, 8, 2, 2, 1, 0, 0, 0, 0, 24, 36, 38, 14, 8, 8, 2, 2, 1, 0, 0, 0, 0, 24, 54, 42, 38, 14, 8, 8, 2, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

Index entries for sequences related to compositions

EXAMPLE

Rows start:

  1;

  0, 1;

  0, 2, 1;

  0, 0, 2, 1;

  0, 0, 2, 2, 1;

  0, 0, 6, 2, 2, 1;

  0, 0, 0, 8, 2, 2, 1;

  0, 0, 0, 6, 8, 2, 2, 1;

  ...

T(7,4)=8 since 7 can be written as 4+3 =4+2+1 =4+1+2 =3+4 =2+4+1 =2+1+4 =1+4+2 =1+2+4.

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1,

      `if`(i<1, [][], zip((x, y)->x+y, [b(n, i-1)],

      `if`(i>n, [], [0, b(n-i, i-1)]), 0)[]))

    end:

T:= proc(n, k) local l; l:= [b(n-k, k-1)];

       add(l[i]*(i)!, i=1..nops(l))

    end:

seq(seq(T(n, k), k=1..n), n=1..20);  # Alois P. Heinz, Nov 20 2012

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, Plus @@ PadRight[{b[n, i-1], If[i>n, {}, Join[{0}, b[n-i, i-1]]]}]]]; T[n_, k_] := Module[{l}, l = b[n-k, k-1]; Sum[l[[i]]*i!, {i, 1, Length[l]}]]; Table[Table [T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-Fran├žois Alcover, Jan 31 2014, after Alois P. Heinz *)

CROSSREFS

Cf. A026836, A072574. Row sums are A032020. Column sums appear to be A001339 (offset). Starting terms of columns tend towards A072576 as k increases.

Sequence in context: A226864 A257399 A168313 * A025872 A280125 A280586

Adjacent sequences:  A072572 A072573 A072574 * A072576 A072577 A072578

KEYWORD

nonn,tabl,changed

AUTHOR

Henry Bottomley, Jun 21 2002

STATUS

approved

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Last modified August 14 19:09 EDT 2020. Contains 336483 sequences. (Running on oeis4.)