cp's OEIS Frontend

A138151 Irregular triangle read by rows in which rows 1..n (when read together) list all the parts in the partitions of n and row n starts with the partitions of n that do not contain 1 as a part (in the order used for A080577).

%I A138151 #46 Jul 30 2025 23:13:55
%S A138151 1,2,1,3,1,1,4,2,2,1,1,1,5,3,2,1,1,1,1,1,6,4,2,3,3,2,2,2,1,1,1,1,1,1,
%T A138151 1,7,5,2,4,3,3,2,2,1,1,1,1,1,1,1,1,1,1,1,8,6,2,5,3,4,4,4,2,2,3,3,2,2,
%U A138151 2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,9,7,2,6,3,5,4,5,2,2,4,3,2,3,3,3,3,2,2
%N A138151 Irregular triangle read by rows in which rows 1..n (when read together) list all the parts in the partitions of n and row n starts with the partitions of n that do not contain 1 as a part (in the order used for A080577).
%C A138151 The remainder of row n is necessarily A000041(n-1) 1's.
%C A138151 Previous name: A shell model of partitions. Row n lists the parts of the last section of the set of partitions of n.
%C A138151 Row n lists the nonzero terms of the n-th row of A138136 together with A000041(n-1) 1's.
%C A138151 Row n is also the n-th row of A138138 in reverse order.
%H A138151 Robert Price, <a href="/A138151/b138151.txt">Table of n, a(n) for n = 1..4630, 20 rows.</a>
%H A138151 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/partit02.jpg">Illustration of the rows 1..6 of the triangle</a>
%e A138151 Triangle begins:
%e A138151 1
%e A138151 2,1
%e A138151 3,1,1
%e A138151 4,2,2,1,1,1
%e A138151 5,3,2,1,1,1,1,1,
%e A138151 6,4,2,3,3,2,2,2,1,1,1,1,1,1,1
%e A138151 7,5,2,4,3,3,2,2,1,1,1,1,1,1,1,1,1,1,1
%e A138151 8,6,2,5,3,4,4,4,2,2,3,3,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%e A138151 9,7,2,6,3,5,4,5,2,2,4,3,2,3,3,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%t A138151 Table[Cases[IntegerPartitions[n], x_ /; Last[x] != 1] ~Join~ConstantArray[{1}, PartitionsP[n - 1]], {n, 8}] // Flatten (* _Robert Price_, May 22 2020 *)
%Y A138151 Mirror of A138138.
%Y A138151 Row lengths give A138137.
%Y A138151 Row sums give A138879.
%Y A138151 Column 1 gives A000027.
%Y A138151 Right border gives A000012.
%Y A138151 Another version is A138121 which is the mirror of A135010.
%Y A138151 Cf. A000041, A080577, A138136, A139094, A139100.
%K A138151 nonn,tabf
%O A138151 1,2
%A A138151 _Omar E. Pol_, Mar 21 2008
%E A138151 New name and comments edited by _Peter Munn_ and _Omar E. Pol_, Jul 25 2025