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%I #11 May 31 2021 05:37:59
%S 1,2,7,27,121,587,2983,15744,85375,473259,2670383,15293119,88686530,
%T 519864702,3075894246,18348407371,110244289384,666651567920,
%U 4054481396896,24786629709850,152241407914480,939069024577371,5815027475345028,36137289604644570
%N Signed partitions of n into n parts in {-n..n} (allowing zero as a part).
%C The normal definition of signed partitions does not allow zero as a part.
%C Signed partitions of n into n parts in {-n..n}\{0}: A211474.
%e a(3) = 7: (-3,3,3), (-2,2,3), (-1,1,3), (-1,2,2), (0,0,3), (0,1,2), (1,1,1).
%p b:= proc(h, i, t, n) option remember;
%p `if`(i=0, `if`(h=0, 1, 0), `if`(h<0 or i*n<h, 0,
%p add (b(h+j, i-1, j, n), j=-n..t)))
%p end:
%p a:= n-> b(n$4):
%p seq (a(n), n=1..15); # _Alois P. Heinz_, Apr 12 2012
%t Table[(IntegerPartitions[n, {1, n}] // Length) + Sum[Sum[(IntegerPartitions[k, {j}, Range[n]] // Length) * (IntegerPartitions[n + k, {1, n - j}, Range[n]] // Length), {j, 0, n - 2}], {k, 1, n*Floor[(n - 1)/2]}], {n, 14}]
%t (* Second program: *)
%t b[h_, i_, t_, n_] := b[h, i, t, n] =
%t If[i == 0, If[h == 0, 1, 0], If[h < 0 || i*n < h, 0,
%t Sum[b[h + j, i - 1, j, n], {j, Range[-n, t]}]]];
%t a[n_] := b[n, n, n, n];
%t Array[a, 24] (* _Jean-François Alcover_, May 31 2021, after _Alois P. Heinz_ *)
%Y Cf. A211474.
%K nonn
%O 1,2
%A _David Scambler_, Apr 12 2012
%E More terms from _Alois P. Heinz_, Apr 12 2012
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