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%I #20 Aug 28 2019 08:51:37
%S 1,1,1,1,1,1,1,2,1,1,2,1,1,3,2,1,1,3,3,1,1,4,4,2,1,1,4,5,3,1,1,5,7,5,
%T 2,1,1,5,8,6,3,1,1,6,10,9,5,2,1,1,6,12,11,7,3,1,1,7,14,15,10,5,2,1,1,
%U 7,16,18,13,7,3,1,1,8,19,23,18,11,5,2,1,1,8,21,27,23,14,7,3,1,1,9,24,34,30
%N Array a(n,m) for number of partitions of n>=2 with m parts having no part 1. Hence m=1..floor(n/2).
%C The row lengths sequence is floor(n/2) = [1,1,2,2,3,3,4,4,...], see A008619(n-1), n>=2.
%C Obtained from the characteristic partition array A145573 by summing in row n>=2 over entries belonging to like parts number m.
%C The column sequences give A000012, A004526, A001399, A001400, A001401, A001402, A026813 for m=1..7.
%H Alois P. Heinz, <a href="/A145574/b145574.txt">Rows n = 2..200, flattened</a>
%H W. Lang and M. Sjodahl, <a href="/A145574/a145574.txt">First 20 rows of the array and row sums.</a>
%F a(n,m) = sum over entries of A145573(n,k) array which belong to partitions with part number m, for m=1..floor(n/2)). Note that partitions with parts number m>floor(n/2) have always at least one part 1.
%F G.f.: Product_{i>=2} 1/(1- y*x^i). - _Geoffrey Critzer_, Sep 23 2012
%e 1;
%e 1;
%e 1, 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 2, 1;
%e 1, 3, 2, 1;
%e 1, 3, 3, 1;
%e 1, 4, 4, 2, 1;
%p b:= proc(n, i, t) option remember; `if`(2*t>n or t*i<n, 0,
%p `if`(n=0, 1, `if`(i<2, 0, b(n, i-1, t) +b(n-i, i, t-1))))
%p end:
%p a:= (n, m)-> b(n, n, m):
%p seq(seq(a(n, m), m=1..iquo(n, 2)), n=2..30); # _Alois P. Heinz_, Oct 18 2012
%t nn=15; f[list_]:=Select[list,#>0&]; p=Product[1/(1-y x^i), {i, 2, nn}]; Drop[Map[f, CoefficientList[Series[p, {x, 0, nn}], {x, y}]], 1]//Grid (* _Geoffrey Critzer_, Sep 23 2012 *)
%o (Sage) # Prints the table; cf. A011973.
%o for n in (2..20): [Partitions(n, length=m, min_part=2).cardinality() for m in (1..n//2)] # _Peter Luschny_, Oct 18 2012
%Y Cf. A145573, A002865 (row sums).
%K nonn,easy,tabf
%O 2,8
%A _Wolfdieter Lang_ and Malin Sjodahl, Mar 06 2009
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