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A352499 Irregular triangle read by rows: T(n,k) is the sum of all parts of the partition of n into consecutive parts that contains 2*k-1 parts, and the first element of the column k is in row A000384(k).

%I A352499 #44 Apr 12 2023 09:23:31
%S A352499 1,2,3,4,5,6,6,7,0,8,0,9,9,10,0,11,0,12,12,13,0,14,0,15,15,15,16,0,0,
%T A352499 17,0,0,18,18,0,19,0,0,20,0,20,21,21,0,22,0,0,23,0,0,24,24,0,25,0,25,
%U A352499 26,0,0,27,27,0,28,0,0,28,29,0,0,0,30,30,30,0,31,0,0,0,32,0,0,0
%N A352499 Irregular triangle read by rows: T(n,k) is the sum of all parts of the partition of n into consecutive parts that contains 2*k-1 parts, and the first element of the column k is in row A000384(k).
%C A352499 This triangle is formed from the odd-indexed columns of the triangle A285891.
%H A352499 Paolo Xausa, <a href="/A352499/b352499.txt">Table of n, a(n) for n = 1..10490</a> (rows 1..800 of triangle, flattened).
%F A352499 T(n,k) = n*A351824(n,k).
%F A352499 T(n,k) = n*[(2*k-1)|n], where 1 <= k <= floor((sqrt(8*n+1)+1)/4) and [] is the Iverson bracket. - _Paolo Xausa_, Apr 12 2023
%e A352499 Triangle begins:
%e A352499    1;
%e A352499    2;
%e A352499    3;
%e A352499    4;
%e A352499    5;
%e A352499    6,  6;
%e A352499    7,  0;
%e A352499    8,  0;
%e A352499    9,  9;
%e A352499   10,  0;
%e A352499   11,  0;
%e A352499   12, 12;
%e A352499   13,  0;
%e A352499   14,  0;
%e A352499   15, 15, 15;
%e A352499   16,  0,  0;
%e A352499   17,  0,  0;
%e A352499   18, 18,  0;
%e A352499   19,  0,  0;
%e A352499   20,  0, 20;
%e A352499   21, 21,  0;
%e A352499   22,  0,  0;
%e A352499   23,  0,  0;
%e A352499   24, 24,  0;
%e A352499   25,  0, 25;
%e A352499   26,  0,  0;
%e A352499   27, 27,  0;
%e A352499   28,  0,  0, 28;
%e A352499   ...
%e A352499 For n = 21 the partitions of 21 into on odd number of consecutive parts are [21] and [8, 7, 6], so T(21,1) = 1 and T(21,2) = 8 + 7 + 6 = 21. There is no partition of 21 into five consecutive parts so T(21,3) = 0.
%t A352499 A352499[rowmax_]:=Table[Boole[Divisible[n,2k-1]]n,{n,rowmax},{k,Floor[(Sqrt[8n+1]+1)/4]}];A352499[50] (* _Paolo Xausa_, Apr 12 2023 *)
%Y A352499 Row sums give A352257.
%Y A352499 Row n has A351846(n) terms.
%Y A352499 The number of nonzero terms in row n equals A082647(n).
%Y A352499 Cf. A000384, A237048, A245579, A285891, A299765, A351824, A352425.
%K A352499 nonn,tabf
%O A352499 1,2
%A A352499 _Omar E. Pol_, Mar 19 2022