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%I #26 Jun 16 2023 03:16:37
%S 1,1,5,17,78,375,1919,10144,55189,306632,1734019,9948977,57790152,
%T 339241199,2009749140,12002162624,72186635028,436913179401,
%U 2659435211566,16270345814930,100002046716732,617227859736748,3824280874554199,23778486784950053,148329560863192846
%N Signed partitions of n into n parts in {-n..n}\{0}.
%C Zero is not allowed as a part.
%C Signed partitions of n into n parts allowing zero as a part: A211475.
%H William J. Keith, <a href="http://dx.doi.org/10.1007/s00026-011-0085-6">A bijective toolkit for signed partitions</a>, Ann. Combinat. 15 (1) (2011) 95-117.
%e a(3) = 5: (1,1,1), (-1,1,3), (-1,2,2), (-2,2,3}, (-3,3,3).
%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} minus {0}))))
%p end:
%p a:= n-> b(n$4):
%p seq(a(n), n=1..15); # _Alois P. Heinz_, Apr 12 2012
%t Table[1 + Sum[Sum[(IntegerPartitions[k, {j}, Range[n]] // Length) * (IntegerPartitions[n + k, {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]~Complement~{0}}]]];
%t a[n_] := b[n, n, n, n];
%t Array[a, 25] (* _Jean-François Alcover_, May 31 2021, after _Alois P. Heinz_ *)
%Y Cf. A211475.
%K nonn
%O 1,3
%A _David Scambler_, Apr 12 2012
%E More terms from _Alois P. Heinz_, Apr 12 2012
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