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Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the twice odd numbers (A016825) interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.
20

%I #46 Nov 05 2024 05:39:55

%S 2,6,10,2,14,0,18,6,22,0,2,26,10,0,30,0,0,34,14,6,38,0,0,2,42,18,0,0,

%T 46,0,10,0,50,22,0,0,54,0,0,6,58,26,14,0,2,62,0,0,0,0,66,30,0,0,0,70,

%U 0,18,10,0,74,34,0,0,0,78,0,0,0,6,82,38,22,0,0,2

%N Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the twice odd numbers (A016825) interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.

%C Gives an identity for the twice sigma function (A074400), the sum of the even divisors of 2n.

%C Alternating sum of row n equals A074400(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = 2*A000203(n) = A074400(n).

%C Row n has length A003056(n) hence the first element of column k is in row A000217(k).

%C The number of positive terms in row n is A001227(n).

%C For more information see A196020.

%F T(n,k) = 2*A196020(n,k).

%e Triangle begins:

%e 2;

%e 6;

%e 10, 2;

%e 14, 0;

%e 18, 6;

%e 22, 0, 2;

%e 26, 10, 0;

%e 30, 0, 0;

%e 34, 14, 6;

%e 38, 0, 0, 2;

%e 42, 18, 0, 0;

%e 46, 0, 10, 0;

%e 50, 22, 0, 0;

%e 54, 0, 0, 6;

%e 58, 26, 14, 0, 2;

%e 62, 0, 0, 0, 0;

%e 66, 30, 0, 0, 0;

%e 70, 0, 18, 10, 0;

%e 74, 34, 0, 0, 0;

%e 78, 0, 0, 0, 6;

%e 82, 38, 22, 0, 0, 2;

%e 86, 0, 0, 14, 0, 0;

%e 90, 42, 0, 0, 0, 0;

%e 94, 0, 26, 0, 0, 0;

%e ...

%e For n = 9 the divisors of 2*9 = 18 are 1, 2, 3, 6, 9, 18, therefore the sum of the even divisors of 18 is 2 + 6 + 18 = 26. On the other hand the 9th row of triangle is 34, 14, 6, therefore the alternating row sum is 34 - 14 + 6 = 26, equaling the sum of the even divisors of 18.

%e If n is even then the alternating sum of the n-th row of triangle is simpler than the sum of the even divisors of 2n. Example: for n = 12 the sum of the even divisors of 2*12 = 24 is 2 + 4 + 6 + 8 + 12 + 24 = 56, and the alternating sum of the 12th row of triangle is 46 - 0 + 10 - 0 = 56.

%Y Cf. A000203, A000217, A001227, A003056, A016825, A074400, A196020, A211343, A228813, A231345, A231347, A235791, A235794, A236104, A236112.

%K nonn,tabf

%O 1,1

%A _Omar E. Pol_, Jan 23 2014