A302247 - OEIS

%I #44 Jun 17 2022 12:00:03

%S 1,1,1,2,1,1,1,1,2,3,1,1,1,1,1,1,1,2,2,2,3,4,1,1,1,1,1,1,1,1,1,1,1,1,

%T 2,2,2,2,3,3,4,5,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,

%U 2,3,3,3,3,4,4,5,6,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

%N Irregular triangle read by rows in which row n lists all parts of all partitions of n, in nondecreasing order.

%C Also due to the correspondence divisor/part row n lists the terms of the n-th row of A338156 in nondecreasing order. In other words: row n lists in nondecreasing order the divisors of the terms of the n-th row of A176206. - _Omar E. Pol_, Jun 16 2022

%H Paolo Xausa, <a href="/A302247/b302247.txt">Table of n, a(n) for n = 1..9687</a>, (rows 1..18 of triangle, flattened)

%e Triangle begins:

%e   1;

%e   1,1,2;

%e   1,1,1,1,2,3;

%e   1,1,1,1,1,1,1,2,2,2,3,4;

%e   1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;

%e   1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6;

%e   ...

%e For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1]. There are seven 1's, three 2's, only one 3 and only one 4, so the 4th row of this triangle is [1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4].

%e On the other hand for n = 4 the 4th row of A176206 is [4, 3, 2, 2, 1, 1, 1] and the divisors of these terms are [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1], [1] the same as the 4th row of A338156. These divisors listed in nondecreasing order give the 4th row of this triangle. - _Omar E. Pol_, Jun 16 2022

%t nrows=10; Array[Sort[Flatten[IntegerPartitions[#]]]&,nrows] (* _Paolo Xausa_, Jun 16 2022 *)

%o (PARI) row(n) = my(list = List()); forpart(p=n, for (k=1, #p, listput(list, p[k]));); vecsort(Vec(list)); \\ _Michel Marcus_, Jun 16 2022

%Y Mirror of A302246.

%Y Row n has length A006128(n).

%Y The sum of row n is A066186(n).

%Y The number of parts k in row n is A066633(n,k).

%Y The sum of all parts k in row n is A138785(n,k).

%Y The number of parts >= k in row n is A181187(n,k).

%Y The sum of all parts >= k in row n is A206561(n,k).

%Y The number of parts <= k in row n is A210947(n,k).

%Y The sum of all parts <= k in row n is A210948(n,k).

%Y First differs from both A026791 and A080576 at a(17) = T(4,7).

%Y Cf. A000041, A027750, A176206, A221529, A336812, A338156.

%K nonn,tabf

%O 1,4

%A _Omar E. Pol_, Apr 05 2018