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A184957 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) is the number of compositions of n into k parts the first of which is >= all the other parts. 4
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 3, 4, 4, 1, 1, 1, 3, 6, 7, 5, 1, 1, 1, 4, 8, 11, 11, 6, 1, 1, 1, 4, 11, 17, 19, 16, 7, 1, 1, 1, 5, 13, 26, 32, 31, 22, 8, 1, 1, 1, 5, 17, 35, 54, 56, 48, 29, 9, 1, 1, 1, 6, 20, 48, 82, 102, 93, 71, 37, 10, 1, 1, 1, 6, 24, 63, 120, 172, 180, 148, 101, 46, 11, 1, 1, 1, 7, 28, 81, 170, 272, 331, 302, 227, 139, 56, 12, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

If the final diagonal is omitted, this gives the triangular array visible in A156041 and A186807.

LINKS

Table of n, a(n) for n=1..105.

FORMULA

T(n,k) = A156041(n-k,k).

E.g.f.:  Sum_{i>=1} x^i/(1 - y*( x - x^(i+1))/(1-x) )/(1-x). - Geoffrey Critzer, Jul 15 2013

EXAMPLE

Triangle begins:

[1],

[1, 1],

[1, 1, 1],

[1, 2, 1, 1],

[1, 2, 3, 1, 1],

[1, 3, 4, 4, 1, 1],

[1, 3, 6, 7, 5, 1, 1],

[1, 4, 8, 11, 11, 6, 1, 1],

[1, 4, 11, 17, 19, 16, 7, 1, 1],

[1, 5, 13, 26, 32, 31, 22, 8, 1, 1],

[1, 5, 17, 35, 54, 56, 48, 29, 9, 1, 1],

...

MAPLE

# The following Maple program is a modification of Alois P. Heinz's program for A156041

b:= proc(n, i, m) option remember;

       if n<0 then 0 elif n=0 then 1 elif i=1 then

      `if`(n<=m, 1, 0) else add(b(n-k, i-1, m), k=0..m) fi

    end:

A:= (n, k)-> add(b(n-m, k-1, m), m=ceil(n/k)..n):

[seq([seq(A(d-k, k), k=1..d)], d=1..14)];

MATHEMATICA

Map[Select[#, #>0&]&, Drop[nn=11; CoefficientList[Series[Sum[x^i/(1-y(x-x^(i+1))/(1-x)), {i, 1, nn}], {x, 0, nn}], {x, y}], 1]]//Grid (* Geoffrey Critzer, Jul 15 2013 *)

CROSSREFS

Cf. A156040, A156041, A186807.

Sequence in context: A046854 A187660 A066170 * A228349 A285718 A205792

Adjacent sequences:  A184954 A184955 A184956 * A184958 A184959 A184960

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Feb 27 2011

STATUS

approved

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Last modified July 24 02:08 EDT 2017. Contains 289717 sequences.