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A266537 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the twice odd numbers (A016825) interleaved with 2*k-1 zeros, and the first positive element of column k is in the row A002378(k), with T(1,1) = 0.

%I A266537 #25 Apr 19 2016 02:40:56
%S A266537 0,2,0,6,0,10,2,0,0,14,0,0,0,18,6,0,0,22,0,2,0,0,0,26,10,0,0,0,0,30,0,
%T A266537 0,0,0,0,34,14,6,0,0,0,38,0,0,2,0,0,0,0,42,18,0,0,0,0,0,0,46,0,10,0,0,
%U A266537 0,0,0,50,22,0,0,0,0,0,0,54,0,0,6,0,0,0,0,58,26,14,0,2
%N A266537 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the twice odd numbers (A016825) interleaved with 2*k-1 zeros, and the first positive element of column k is in the row A002378(k), with T(1,1) = 0.
%C A266537 Gives an identity for A146076. Alternating sum in row n equals the sum of even divisors of n.
%C A266537 Even-indexed rows of the triangle give A236106.
%C A266537 If T(n,k) = 6 then T(n+2,k+1) = 2, the first element of the column k+1.
%F A266537 T(n,k) = 0, if n is odd.
%F A266537 T(n,k) = 2*A196020(n/2,k) = A236106(n/2,k), if n is even.
%e A266537 Triangle begins:
%e A266537 0;
%e A266537 2;
%e A266537 0;
%e A266537 6;
%e A266537 0;
%e A266537 10,  2;
%e A266537 0,   0;
%e A266537 14,  0;
%e A266537 0,   0;
%e A266537 18,  6;
%e A266537 0,   0;
%e A266537 22,  0,  2;
%e A266537 0,   0,  0;
%e A266537 26, 10,  0;
%e A266537 0,   0,  0;
%e A266537 30,  0,  0;
%e A266537 0,   0,  0;
%e A266537 34, 14,  6;
%e A266537 0,   0,  0;
%e A266537 38,  0,  0,  2;
%e A266537 0,   0,  0,  0;
%e A266537 42, 18,  0,  0;
%e A266537 0,   0,  0,  0;
%e A266537 46,  0, 10,  0;
%e A266537 0,   0,  0,  0;
%e A266537 50, 22,  0,  0;
%e A266537 0,   0,  0,  0;
%e A266537 54,  0,  0,  6;
%e A266537 0,   0,  0,  0;
%e A266537 58, 26, 14,  0,  2;
%e A266537 ...
%e A266537 For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12 and the sum of even divisors of 12 is 2 + 4 + 6 + 12 = 24. On the other hand, the 12th row of the triangle is 22, 0, 2, so the alternating row sum is 22 - 0 + 2 = 24, equaling the sum of even divisors of 12.
%Y A266537 Cf. A002378, A016825, A146076, A196020, A236106, A237593, A271343.
%K A266537 nonn,tabf
%O A266537 1,2
%A A266537 _Omar E. Pol_, Apr 05 2016