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
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Sep 12 2012
STATUS
approved
