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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.

%I A072576 #36 Nov 18 2019 07:06:31
%S A072576 1,2,2,8,8,14,38,44,68,98,242,272,440,590,878,1772,2180,3194,4466,
%T A072576 6320,8432,16190,19262,28580,38276,54314,70730,99152,163328,204230,
%U A072576 286670,386132,527132,695978,941738,1220984,1950128,2390294,3321398,4342148,5929532,7616642,10284410
%N 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.
%C A072576 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
%H A072576 Seiichi Manyama, <a href="/A072576/b072576.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%H A072576 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A072576 a(n) = Sum_k (k+1)! * A060016(n,k) = Sum_k (k+1) * A072574(n,k).
%F A072576 a(n) = Sum_k (k+1)! * A008289(n,k). - _Alois P. Heinz_, Dec 12 2012
%e A072576 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.
%e A072576 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
%p A072576 b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
%p A072576       -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0))) end:
%p A072576 a:= proc(n) local l; l:= b(n, n): add( i! * l[i], i=1..nops(l)) end:
%p A072576 seq(a(n), n=0..50);  # _Alois P. Heinz_, Dec 12 2012
%t A072576 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_ *)
%o A072576 (PARI)
%o A072576 N=66;  q='q+O('q^N);
%o A072576 gf=sum(n=0,N, (n+1)!*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
%o A072576 Vec(gf)
%o A072576 /* _Joerg Arndt_, Oct 20 2012 */
%Y A072576 Cf. A072575.
%Y A072576 Cf. A032020. - _Alois P. Heinz_, Dec 12 2012
%K A072576 nonn
%O A072576 0,2
%A A072576 _Henry Bottomley_, Jun 21 2002