cp's OEIS Frontend

A354009 Irregular triangle read by rows in which row n lists the partitions of n into an odd number of equal parts, in nonincreasing order.

%I A354009 #93 Jul 16 2022 17:16:30
%S A354009 1,2,3,1,1,1,4,5,1,1,1,1,1,6,2,2,2,7,1,1,1,1,1,1,1,8,9,3,3,3,1,1,1,1,
%T A354009 1,1,1,1,1,10,2,2,2,2,2,11,1,1,1,1,1,1,1,1,1,1,1,12,4,4,4,13,1,1,1,1,
%U A354009 1,1,1,1,1,1,1,1,1,14,2,2,2,2,2,2,2,15,5,5,5,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A354009 Irregular triangle read by rows in which row n lists the partitions of n into an odd number of equal parts, in nonincreasing order.
%C A354009 The number of partitions in row n equals A001227(n), the number of odd divisors of n, and equals the number of partitions of n into consecutive parts, and equals the number of subparts in the symmetric representation of sigma(n).
%C A354009 The sum of row n equals A245579(n), the sum of all parts of all partitions of n into consecutive parts.
%C A354009 The length of row n equals A000593(n), the sum of the odd divisors of n.
%C A354009 Row n has length 1 if and only if n is a power of 2.
%C A354009 Is the right border the same as A006519?
%e A354009 Triangle begins:
%e A354009    [1];
%e A354009    [2];
%e A354009    [3], [1,1,1];
%e A354009    [4];
%e A354009    [5], [1,1,1,1,1];
%e A354009    [6], [2,2,2];
%e A354009    [7], [1,1,1,1,1,1,1];
%e A354009    [8];
%e A354009    [9], [3,3,3], [1,1,1,1,1,1,1,1,1];
%e A354009   [10], [2,2,2,2,2];
%e A354009   [11], [1,1,1,1,1,1,1,1,1,1,1];
%e A354009   [12], [4,4,4];
%e A354009   [13], [1,1,1,1,1,1,1,1,1,1,1,1,1];
%e A354009   [14], [2,2,2,2,2,2,2];
%e A354009   [15], [5,5,5], [3,3,3,3,3], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1];
%e A354009   [16];
%e A354009   ...
%e A354009 For n = 10, in row 10 there are two partitions, equaling the number of odd divisors of 10, they are [1, 5], and equaling the number of partitions of 10 into consecutive parts, they are [10], [4, 3, 2, 1], and equaling the number of subparts in the symmetric representation of sigma(10), they are [9, 9].
%e A354009 The sum of row 10 is [10] + [2 + 2 + 2 + 2 + 2] = 20 equaling the sum of all parts of all partitions of 10 into consecutive parts, that is [10] + [4 + 3 + 2 + 1] = 20.
%e A354009 The length of row 10 is equal to 6 equaling the sum of the odd divisors of 10, that is 1 + 5 = 6.
%t A354009 Table[ConstantArray[n/#, #] & /@ Select[Divisors[n], OddQ], {n, 15}] // Flatten (* _Michael De Vlieger_, Jul 15 2022 *)
%o A354009 (PARI) row(n) = my(v=[]); fordiv(n, d, if ((n/d)%2, v = concat(v, vector(n/d, k, d)))); Vecrev(v); \\ _Michel Marcus_, Jul 16 2022
%Y A354009 Subsequence of A244051.
%Y A354009 The number of partitions in row n equals A001227(n).
%Y A354009 Row lengths give A000593.
%Y A354009 Row sums give A245579.
%Y A354009 Column 1 gives A000027.
%Y A354009 Cf. A000079, A000203, A006519, A237593, A279387 (subparts), A299765.
%K A354009 nonn,tabf
%O A354009 1,2
%A A354009 _Omar E. Pol_, Jul 13 2022