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A145975 Triangle read by rows, partition triangle A027293 convolved with A010815.

%I A145975 #15 Jun 16 2020 02:02:12
%S A145975 1,1,-1,2,-1,-1,3,-2,-1,0,5,-3,-2,0,0,7,-5,-3,0,0,1,11,-7,-5,0,0,1,0,
%T A145975 15,-11,-7,0,0,2,0,1,22,-15,-11,0,0,3,0,1,0,30,-22,-15,0,0,5,0,2,0,0,
%U A145975 42,-30,-22,0,0,7,0,3,0,0,0,56,-42,-30,0,0,11,0,5,0,0,0,0
%N A145975 Triangle read by rows, partition triangle A027293 convolved with A010815.
%C A145975 Row sums = [1, 0, 0, 0,...]. (a set of matrix operations equivalent to the comment in A010815 that "convolved with the partition numbers = [1, 0, 0, 0,...].
%H A145975 Robert Price, <a href="/A145975/b145975.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%F A145975 Triangle read by rows, = (A027293 * (A010815 * 0^(n-k)); 0<=k<=n. A027293 = an infinite lower triangular matrix with A000041 in every column (the partition numbers). A010815 = (1, -1, -1, 0, 0, 1,...)
%e A145975 First few rows of the triangle =
%e A145975 1;
%e A145975 1, -1;
%e A145975 2, -1, -1;
%e A145975 3, -2, -1, 0;
%e A145975 5, -3, -2, 0, 0;
%e A145975 7, -5, -3, 0, 0, 1;
%e A145975 11, -7, -5, 0, 0, 1, 0;
%e A145975 15, -11, -7, 0, 0, 2, 0, 1;
%e A145975 22, -15, -11, 0, 0, 3, 0, 1, 0;
%e A145975 30, -22, -15, 0, 0, 5, 0, 2, 0, 0;
%e A145975 42, -30, -22, 0, 0, 7, 0, 3, 0, 0, 0;
%e A145975 ...
%t A145975 Table[Count[Flatten[Union /@ IntegerPartitions@n],k]*SeriesCoefficient[Product[1 - x^i, {i, k - 1}], {x, 0, k - 1}], {n, 12}, {k, n}] // Flatten (* _Robert Price_, Jun 15 2020 *)
%Y A145975 A027293, Cf. A010815, A000041
%K A145975 tabl,sign
%O A145975 1,4
%A A145975 _Gary W. Adamson_, Oct 25 2008
%E A145975 Missing zero at a(55) inserted by _Robert Price_, Jun 15 2020