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A231345 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the odd numbers interleaved with k-1 zeros but T(n,1) = -1 and the first element of column k is in row k(k+1)/2.

%I A231345 #55 Feb 16 2025 08:33:20
%S A231345 -1,-1,-1,1,-1,0,-1,3,-1,0,1,-1,5,0,-1,0,0,-1,7,3,-1,0,0,1,-1,9,0,0,
%T A231345 -1,0,5,0,-1,11,0,0,-1,0,0,3,-1,13,7,0,1,-1,0,0,0,0,-1,15,0,0,0,-1,0,
%U A231345 9,5,0,-1,17,0,0,0,-1,0,0,0,3,-1,19,11,0,0,1
%N A231345 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the odd numbers interleaved with k-1 zeros but T(n,1) = -1 and the first element of column k is in row k(k+1)/2.
%C A231345 Gives an identity for the abundance of n. Alternating sum of row n equals the abundance of n, i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A033880(n).
%C A231345 Row n has length A003056(n) hence the first element of column k is in row A000217(k).
%H A231345 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Abundance.html">Abundance</a>
%H A231345 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuasiperfectNumber.html">Quasiperfect Number</a>
%F A231345 T(n,1) = -1; T(n,k) = A196020(n,k), for k >= 2.
%e A231345 Triangle begins:
%e A231345   -1;
%e A231345   -1;
%e A231345   -1,  1;
%e A231345   -1,  0;
%e A231345   -1,  3;
%e A231345   -1,  0,  1;
%e A231345   -1,  5,  0;
%e A231345   -1,  0,  0;
%e A231345   -1,  7,  3;
%e A231345   -1,  0,  0,  1;
%e A231345   -1,  9,  0,  0;
%e A231345   -1,  0,  5,  0;
%e A231345   -1, 11,  0,  0;
%e A231345   -1,  0,  0,  3;
%e A231345   -1, 13,  7,  0,  1;
%e A231345   -1,  0,  0,  0,  0;
%e A231345   -1, 15,  0,  0,  0;
%e A231345   -1,  0,  9,  5,  0;
%e A231345   -1, 17,  0,  0,  0;
%e A231345   -1,  0,  0,  0,  3;
%e A231345   -1, 19, 11,  0,  0,  1;
%e A231345   -1,  0,  0,  7,  0,  0;
%e A231345   -1, 21,  0,  0,  0,  0;
%e A231345   -1,  0, 13,  0,  0,  0;
%e A231345   ...
%e A231345 For n = 15 the divisors of 15 are 1, 3, 5, 15 hence the abundance of 15 is 1 + 3 + 5 + 15 - 2*15 = 1 + 3 + 5 - 15 = -6. On the other hand the 15th row of triangle is -1, 13, 7, 0, 1, hence the alternating row sum is -1 - 13 + 7 - 0 + 1 = -6, equalling the abundance of 15.
%e A231345 If n is even then the alternating sum of the n-th row of triangle is simpler than the sum of divisors of n minus 2*n. Example: the sum of divisors of 24 minus 2*24 is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 - 2*24 = 60 - 48 = 12, and the alternating sum of the 24th row of triangle is -1 - 0 + 13 - 0 + 0 - 0 = 12.
%Y A231345 Column 2 is A193356.
%Y A231345 Cf. A000203, A000217, A000396, A003056, A005100, A005843, A033879, A033880, A069283, A196020, A212119, A228813, A231347, A235791, A235794, A236104, A236106, A236112, A237593.
%K A231345 sign,tabf,nice
%O A231345 1,8
%A A231345 _Omar E. Pol_, Dec 26 2013