A236106 - OEIS

%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