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A216652 Triangular array read by rows: T(n,k) is the number of compositions of n into exactly k distinct parts. 10
1, 1, 1, 2, 1, 2, 1, 4, 1, 4, 6, 1, 6, 6, 1, 6, 12, 1, 8, 18, 1, 8, 24, 24, 1, 10, 30, 24, 1, 10, 42, 48, 1, 12, 48, 72, 1, 12, 60, 120, 1, 14, 72, 144, 120, 1, 14, 84, 216, 120, 1, 16, 96, 264, 240, 1, 16, 114, 360, 360, 1, 18, 126, 432, 600, 1, 18, 144, 552, 840 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Same as A072574, with zeros dropped. [Joerg Arndt, Oct 20 2012]

Row sums = A032020.

Row n contains A003056(n) = floor((sqrt(8*n+1)-1)/2) terms (number of terms increases by one at each triangular number).

LINKS

Alois P. Heinz, Rows n = 1..500, flattened

B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97.

FORMULA

G.f.: Sum_{i>=0} Product_{j=1..i} y*j*x^j/(1-x^j).

T(n,k) = A008289(n,k)*k!.

EXAMPLE

Triangle starts:

[ 1]  1;

[ 2]  1;

[ 3]  1, 2;

[ 4]  1, 2;

[ 5]  1, 4;

[ 6]  1, 4, 6;

[ 7]  1, 6, 6;

[ 8]  1, 6, 12;

[ 9]  1, 8, 18;

[10]  1, 8, 24, 24;

[11]  1, 10, 30, 24;

[12]  1, 10, 42, 48;

[13]  1, 12, 48, 72;

[14]  1, 12, 60, 120;

[15]  1, 14, 72, 144, 120;

[16]  1, 14, 84, 216, 120;

[17]  1, 16, 96, 264, 240;

[18]  1, 16, 114, 360, 360;

[19]  1, 18, 126, 432, 600;

[20]  1, 18, 144, 552, 840;

T(5,2) = 4 because we have: 4+1, 1+4, 3+2, 2+3.

MAPLE

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

      `if`(k<1, 0, b(n, k-1) +b(n-k, k))))

    end:

T:= (n, k)-> b(n-k*(k+1)/2, k)*k!:

seq(seq(T(n, k), k=1..floor((sqrt(8*n+1)-1)/2)), n=1..24);  # Alois P. Heinz, Sep 12 2012

MATHEMATICA

nn=20; f[list_]:=Select[list, #>0&]; Map[f, Drop[CoefficientList[Series[ Sum[Product[j y x^j/(1-x^j), {j, 1, k}], {k, 0, nn}], {x, 0, nn}], {x, y}], 1]]//Flatten

CROSSREFS

Cf. A003056, A008289, A072574, A097910.

Sequence in context: A343411 A287477 A231473 * A331980 A055684 A300584

Adjacent sequences:  A216649 A216650 A216651 * A216653 A216654 A216655

KEYWORD

nonn,tabf

AUTHOR

Geoffrey Critzer, Sep 12 2012

STATUS

approved

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Last modified July 29 17:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)