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A100568 Number of compositions of n(n^2+1)/2 into n distinct parts each no more than n^2.

%I A100568 #20 Oct 31 2018 22:06:19
%S A100568 1,1,4,48,2064,167280,23136480,4824953280,1417422988800,
%T A100568 557894688341760,283527366696806400,180770613278509900800,
%U A100568 141310830114906688051200,132919668653581764822067200,148111929489204170921816985600,192952383265326280925512415232000
%N A100568 Number of compositions of n(n^2+1)/2 into n distinct parts each no more than n^2.
%C A100568 In an n X n magic square, each row and column is a composition of type described.
%H A100568 Alois P. Heinz, <a href="/A100568/b100568.txt">Table of n, a(n) for n = 0..150</a>
%H A100568 Henry Bottomley, <a href="http://se16.info/js/partitions.htm">Partition and composition calculator</a>
%F A100568 a(n) = A000142(n)*A052456(n). a(n) is close to n^(2n-5/2)*sqrt(6/(pi*e)) in the sense that the ratio between the two tends to 1 as n increases. Experimentally, something like n^(2n) * sqrt(6 / (pi * e * (n^5 - 1.366...n^4 + 1.146...n^3 - 0.826...n^2 + 0.413...n + 0.115...))) seems to be even closer.
%e A100568 a(2)=4 since 5 can be written 1+4, 2+3, 3+2 or 4+1.
%p A100568 b := proc(n, i, t) option remember;
%p A100568 `if`(n<t*(t+1)/2 or n>t*(2*i-t+1)/2, 0,
%p A100568 `if`(n=0, 1, b(n, i-1, t) + `if`(n<i, 0, b(n-i, i-1, t-1)))) end:
%p A100568 a := n -> `if`(n=0, 1, n!*b(n*(n^2+1)/2, n^2, n)): seq(a(n), n=0..12); # _Peter Luschny_, May 06 2014, after _Alois P. Heinz_
%t A100568 RecursionLimit = 1000; b[n_, i_, t_] /; i < t || n < t*((t+1)/2) || n > t*((2*i-t+1)/2) = 0; b[0, _, _] = 1; b[n_, i_, t_] :=  b[n, i, t] = b[n, i-1, t] + If[n < i, 0, b[n-i, i-1, t-1]]; a[_, 0] = 1; a[0, _] = 0; a[n_, k_] :=  With[{s = k*(k*n+1)}, If[Mod[s, 2] == 1, 0, b[s/2, k*n, k]]]; a[n_] := a[n] = a[n, n]*n!; Table[Print[a[n]];  a[n], {n, 0,   14}] (* _Jean-François Alcover_, Aug 15 2013, after _Alois P. Heinz_ *)
%Y A100568 Cf. A000142, A052456.
%K A100568 nonn
%O A100568 0,3
%A A100568 _Henry Bottomley_, Nov 28 2004