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A072576 Limit of number of compositions (ordered partitions) of m into distinct parts where largest part is exactly m-n, for m sufficiently large given n. 5
1, 2, 2, 8, 8, 14, 38, 44, 68, 98, 242, 272, 440, 590, 878, 1772, 2180, 3194, 4466, 6320, 8432, 16190, 19262, 28580, 38276, 54314, 70730, 99152, 163328, 204230, 286670, 386132, 527132, 695978, 941738, 1220984, 1950128, 2390294, 3321398, 4342148, 5929532, 7616642, 10284410 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Consider an ordered 1 X n tiling of white tiles whose lengths are all distinct from each other, and whose sum is n. Now introduce into this tiling a red square. The resulting number of compositions is a(n). - Gregory L. Simay, Oct 25 2019

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Index entries for sequences related to compositions

FORMULA

a(n) = Sum_k (k+1)! * A060016(n,k) = Sum_k (k+1) * A072574(n,k).

a(n) = Sum_k (k+1)! * A008289(n,k). - Alois P. Heinz, Dec 12 2012

EXAMPLE

a(3) = 8 because for any m > 6 the number of compositions of e.g. m=7 into distinct parts where the largest part is exactly m-3 = 7-3 = 4 is eight, 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.

Note that in the example immediately above, 4 corresponds to the red square, since it is greater than--and therefore distinct from--parts 1,2 and 3, which correspond to the distinct white tiles. More generally, for the compositions of n having all parts distinct, the red square must correspond to a positive integer > n in order for the number of resulting compositions to be a(n). - Gregory L. Simay, Oct 25 2019

MAPLE

b:= proc(n, i) b(n, i):= `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:

a:= proc(n) local l; l:= b(n, n): add( i! * l[i], i=1..nops(l)) end:

seq(a(n), n=0..50);  # Alois P. Heinz, Dec 12 2012

MATHEMATICA

b[n_, i_] := If[n == 0, {1}, If[i<1, {}, Plus @@ PadRight[{b[n, i-1], If[i>n, {}, Prepend[b[n-i, i-1], 0]]}]]]; a[n_] := Module[{l}, l = b[n, n]; Sum[i!*l[[i]], {i, 1, Length[l]}]]; Table[a[n], {n, 0, 50}] (* Jean-Fran├žois Alcover, Jan 31 2014, after Alois P. Heinz *)

PROG

(PARI)

N=66;  q='q+O('q^N);

gf=sum(n=0, N, (n+1)!*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) );

Vec(gf)

/* Joerg Arndt, Oct 20 2012 */

CROSSREFS

Cf. A072575.

Cf. A032020. - Alois P. Heinz, Dec 12 2012

Sequence in context: A151924 A268342 A058524 * A271342 A060818 A082887

Adjacent sequences:  A072573 A072574 A072575 * A072577 A072578 A072579

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jun 21 2002

STATUS

approved

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Last modified September 21 19:57 EDT 2020. Contains 337273 sequences. (Running on oeis4.)