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A235794 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k starts with k zeros and then lists the odd numbers interleaved with k zeros, and the first element of column k is in row k(k+1)/2.

%I A235794 #24 Nov 05 2024 05:39:47
%S A235794 0,1,0,0,3,0,0,1,5,0,0,0,0,0,7,3,0,0,0,1,9,0,0,0,0,5,0,0,11,0,0,0,0,0,
%T A235794 3,0,13,7,0,1,0,0,0,0,0,15,0,0,0,0,0,9,5,0,0,17,0,0,0,0,0,0,0,3,0,19,
%U A235794 11,0,0,1,0,0,7,0,0,0,21,0,0,0,0,0,0
%N A235794 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k starts with k zeros and then lists the odd numbers interleaved with k zeros, and the first element of column k is in row k(k+1)/2.
%C A235794 It appears that the alternating row sums give A120444, the first differences of A004125, i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A120444(n).
%C A235794 Row n has length A003056(n) hence the first element of column k is in row A000217(k).
%e A235794 Triangle begins:
%e A235794   0;
%e A235794   1;
%e A235794   0,  0;
%e A235794   3,  0;
%e A235794   0,  1;
%e A235794   5,  0,  0;
%e A235794   0,  0,  0;
%e A235794   7,  3,  0;
%e A235794   0,  0,  1;
%e A235794   9,  0,  0,  0;
%e A235794   0,  5,  0,  0;
%e A235794   11, 0,  0,  0;
%e A235794   0,  0,  3,  0;
%e A235794   13, 7,  0,  1;
%e A235794   0,  0,  0,  0,  0;
%e A235794   15, 0,  0,  0,  0;
%e A235794   0,  9,  5,  0,  0;
%e A235794   17, 0,  0,  0,  0;
%e A235794   0,  0,  0,  3,  0;
%e A235794   19, 11, 0,  0,  1;
%e A235794   0,  0,  7,  0,  0,  0;
%e A235794   21, 0,  0,  0,  0,  0;
%e A235794   0,  13, 0,  0,  0,  0;
%e A235794   23, 0,  0,  5,  0,  0;
%e A235794   ...
%e A235794 For n = 14 the 14th row of triangle is 13, 7, 0, 1, and the alternating sum is 13 - 7 + 0 - 1 = 5, the same as A120444(14) = 5.
%Y A235794 Cf. A000203, A000217, A003056, A004125, A120444, A196020, A211343, A228813, A231345, A231347, A235791, A236104, A236106, A236112.
%K A235794 nonn,tabf
%O A235794 1,5
%A A235794 _Omar E. Pol_, Jan 23 2014