A239313 - OEIS

%I #20 Apr 20 2016 00:33:09

%S 0,0,1,1,2,0,3,3,4,0,1,5,5,0,6,0,0,7,7,3,8,0,0,1,9,9,0,0,10,0,5,0,11,

%T 11,0,0,12,0,0,3,13,13,7,0,1,14,0,0,0,0,15,15,0,0,0,16,0,9,5,0,17,17,

%U 0,0,0,18,0,0,0,3,19,19,11,0,0,1,20,0,0,7,0,0

%N Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the odd numbers interleaved with k-1 zeros, except the first column which lists 0 together with the nonnegative integers, and the first element of column k is in row k*(k+1)/2.

%C Alternating row sums give the Chowla's function, i.e., sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A048050(n).

%C Row n has length A003056(n) hence column k starts in row A000217(k).

%C Column 1 gives 0 together with A001477.

%C Column 2 is A193356.

%C The number of positive terms in row n is A001227(n), if n >= 3. - _Omar E. Pol_, Apr 18 2016

%F T(n,k) = A196020(n,k), if k >= 2. - _Omar E. Pol_, Apr 18 2016

%e Triangle begins (row n = 1..24):

%e 0;

%e 0;

%e 1,   1;

%e 2,   0;

%e 3,   3;

%e 4,   0,  1;

%e 5,   5,  0;

%e 6,   0,  0;

%e 7,   7,  3;

%e 8,   0,  0,  1;

%e 9,   9,  0,  0;

%e 10,  0,  5,  0;

%e 11, 11,  0,  0;

%e 12,  0,  0,  3;

%e 13, 13,  7,  0,  1;

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

%e 15, 15,  0,  0,  0;

%e 16,  0,  9,  5,  0;

%e 17, 17,  0,  0,  0;

%e 18,  0,  0,  0,  3;

%e 19, 19, 11,  0,  0,  1;

%e 20,  0,  0,  7,  0,  0;

%e 21, 21,  0,  0,  0,  0;

%e 22,  0, 13,  0,  0,  0;

%e ...

%e For n = 15 the divisors of 15 are 1, 3, 5, 15 therefore the sum of divisors of 15 except 1 and 15 is 3 + 5 = 8. On the other hand the 15th row of triangle is 13, 13, 7, 0, 1, hence the alternating row sum is 13 - 13 + 7 - 0 + 1 = 8, equalling the sum of divisors of 15 except 1 and 15.

%e If n is even then the alternating sum of the n-th row of triangle is simpler than the sum of divisors of n, except 1 and n. Example: the sum of divisors of 24 except 1 and 24 is 2 + 3 + 4 + 6 + 8 + 12 = 35, and the alternating sum of the 24th row of triangle is 22 - 0 + 13 - 0 + 0 - 0 = 35.

%Y Cf. A000203, A000217, A001065, A001227, A001477, A193356, A196020, A211343, A212119, A228813, A231345, A231347, A235791, A235794, A236104, A236106, A236112.

%K nonn,tabf

%O 1,5

%A _Omar E. Pol_, Mar 15 2014