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.

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

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-François 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: A343984 A268342 A058524 * A271342 A060818 A082887

Adjacent sequences: A072573 A072574 A072575 * A072577 A072578 A072579

Keyword

nonn

Author

Henry Bottomley, Jun 21 2002

Status

approved