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A130540 Triangle read by rows T(n,k) in which column k lists the terms of A000203 interspersed with (k-1) zeros, 1 <= k <= n.

%I A130540 #23 Apr 19 2020 12:46:26
%S A130540 1,3,1,4,0,1,7,3,0,1,6,0,0,0,1,12,4,3,0,0,1,8,0,0,0,0,0,1,15,7,0,3,0,
%T A130540 0,0,1,13,0,4,0,0,0,0,0,1,18,6,0,0,3,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,1,
%U A130540 28,12,7,4,0,3,0,0,0,0,0,1,14,0,0,0,0,0,0,0,0,0,0,0,1,24,8,0,0,0,0,3,0,0,0,0,0,0,1
%N A130540 Triangle read by rows T(n,k) in which column k lists the terms of A000203 interspersed with (k-1) zeros, 1 <= k <= n.
%C A130540 The original definition was: A127093 * A125093^(-1).
%C A130540 Left border = A000203, sigma(n): (1, 3, 4, 7, 6, ...). Row sums = A007429: (1, 4, 5, 11, 7, 20, 9, ...); = inverse Moebius transform applied to sigma(n); (i.e., inverse Moebius transform applied twice to natural numbers).
%C A130540 T(n,k) is the total number of parts congruent to 0 mod k in the partitions of n into equal parts. - _Omar E. Pol_, Nov 19 2019
%C A130540 From _Omar E. Pol_, Jan 01 2020: (Start)
%C A130540 Conjecture 1: the sum of odd-indexed terms in row n equals A327096(n).
%C A130540 Conjecture 2: the sum of even-indexed terms in row n equals the n-th term of the sequence formed by A000004 and A007429 interleaved.
%C A130540 Conjecture 3: alternating row sums give A288417. (End)
%F A130540 A127093 * A125093^(-1), as infinite lower triangular matrices.
%e A130540 First few rows of the triangle are:
%e A130540    1;
%e A130540    3,  1;
%e A130540    4,  0, 1;
%e A130540    7,  3, 0, 1;
%e A130540    6,  0, 0, 0, 1;
%e A130540   12,  4, 3, 0, 0, 1;
%e A130540    8,  0, 0, 0, 0, 0, 1;
%e A130540   15,  7, 0, 3, 0, 0, 0, 1;
%e A130540   13,  0, 4, 0, 0, 0, 0, 0, 1;
%e A130540   18,  6, 0, 0, 3, 0, 0, 0, 0, 1;
%e A130540   12,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A130540   28, 12, 7, 4, 0, 3, 0, 0, 0, 0, 0, 1;
%e A130540   14,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A130540   24,  8, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1;
%e A130540   24,  0, 6, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A130540   31, 15, 0, 7, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1;
%e A130540 ...
%e A130540 Extended by _Omar E. Pol_, Nov 19 2019
%Y A130540 Cf. A000203, A007429, A127093, A125093, A244051, A288417, A327096.
%K A130540 nonn,tabl
%O A130540 1,2
%A A130540 _Gary W. Adamson_, Jun 03 2007
%E A130540 New name and more terms from _Omar E. Pol_, Nov 19 2019