This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A325350 #11 Mar 03 2021 19:28:36
%S A325350 1,1,2,3,4,6,8,10,13,17,21,26,32,38,46,56,66,78,92,106,124,145,166,
%T A325350 191,220,249,284,325,366,413,468,523,586,659,733,817,913,1011,1121,
%U A325350 1245,1373,1515,1674,1838,2020,2223,2433,2664,2920,3184,3476,3797,4129,4492
%N A325350 Number of integer partitions of n whose augmented differences are weakly decreasing.
%C A325350 The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
%C A325350 The Heinz numbers of these partitions are given by A325389.
%H A325350 Fausto A. C. Cariboni, <a href="/A325350/b325350.txt">Table of n, a(n) for n = 0..500</a>
%F A325350 G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(j*(j+1)/2)) (conjecture). - _Ilya Gutkovskiy_, Apr 25 2019
%e A325350 The a(1) = 1 through a(8) = 13 partitions:
%e A325350 (1) (2) (3) (4) (5) (6) (7) (8)
%e A325350 (11) (21) (31) (32) (42) (52) (53)
%e A325350 (111) (211) (41) (51) (61) (62)
%e A325350 (1111) (311) (321) (421) (71)
%e A325350 (2111) (411) (511) (521)
%e A325350 (11111) (3111) (3211) (611)
%e A325350 (21111) (4111) (4211)
%e A325350 (111111) (31111) (5111)
%e A325350 (211111) (32111)
%e A325350 (1111111) (41111)
%e A325350 (311111)
%e A325350 (2111111)
%e A325350 (11111111)
%e A325350 For example, (4,2,1,1) has augmented differences (3,2,1,1), which are weakly decreasing, so (4,2,1,1) is counted under a(8).
%t A325350 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A325350 Table[Length[Select[IntegerPartitions[n],OrderedQ[Reverse[aug[#]]]&]],{n,0,30}]
%Y A325350 Cf. A007294, A098859, A240026, A320466, A320509, A325349, A325353, A325354, A325356, A325357, A325358, A325361, A325364.
%K A325350 nonn
%O A325350 0,3
%A A325350 _Gus Wiseman_, Apr 23 2019