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%I #16 Feb 21 2024 08:23:47
%S 1,1,1,1,2,1,2,1,1,3,2,1,4,2,1,1,5,3,2,1,6,4,2,1,1,8,5,3,2,1,10,6,4,2,
%T 1,1,12,8,5,3,2,1,15,10,6,4,2,1,1,18,12,8,5,3,2,1,22,15,10,6,4,2,1,1,
%U 27,18,12,8,5,3,2,1,32,22,15,10,6,4,2,1,1
%N Triangle by columns, A000009 in every column shifted down twice for k > 0.
%C Row sums = A038348.
%C Let the triangle = M. Limit_{n->oo} M^n = the partition numbers, A000041;
%C equivalent to the statement A000009(x) = A000041(x)/A000041(x^2), or
%C (1 + x + x^2 + 2x^3 + 2x^4 +3x^5 + 4x^6 + ...) = (1 + x + 2x^2 + 3x^3 + ...)/(1 + x^2 + 2x^4 + 3x^6 + 5x^8 + 7x^10 + ...).
%H Paolo Xausa, <a href="/A173305/b173305.txt">Table of n, a(n) for n = 0..10301</a> (rows 0..200 of the triangle, flattened).
%F Triangle by columns, A000009 in every column shifted down twice for k > 0.
%F T(n,k) = A000009(n-2*k). - _Paolo Xausa_, Feb 21 2024
%e Triangle begins:
%e 1;
%e 1;
%e 1, 1;
%e 2, 1;
%e 2, 1, 1;
%e 3, 2, 1;
%e 4, 2, 1, 1;
%e 5, 3, 2, 1;
%e 6, 4, 2, 1, 1;
%e 8, 5, 3, 2, 1;
%e 10, 6, 4, 2, 1, 1;
%e 12, 8, 5, 3, 2, 1;
%e 15, 10, 6, 4, 2, 1, 1;
%e 18, 12, 8, 5, 3, 2, 1;
%e 22, 15, 10, 6, 4, 2, 1, 1;
%e 27, 18, 12, 8, 5, 3, 2, 1;
%e 32, 22, 15, 10, 6, 4, 2, 1, 1;
%e 38, 27, 18, 12, 8, 5, 3, 2, 1;
%e 46, 32, 22, 15, 10, 6, 4, 2, 1, 1;
%e 54, 38, 27, 18, 12, 8, 5, 3, 2, 1;
%e ...
%t Table[PartitionsQ[n-2*k], {n, 0, 15}, {k, 0, n/2}] (* _Paolo Xausa_, Feb 21 2024 *)
%Y Cf. A173306, A038348, A000009, A000041.
%K nonn,tabf
%O 0,5
%A _Gary W. Adamson_, Feb 15 2010
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