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 A325165 #9 Jan 19 2023 12:31:02
%S A325165 1,0,1,0,0,2,0,0,0,3,0,1,0,0,4,0,2,0,0,0,5,0,3,2,0,0,0,6,0,4,4,0,0,0,
%T A325165 0,7,0,5,6,3,0,0,0,0,8,0,7,8,6,0,0,0,0,0,9,0,9,10,9,4,0,0,0,0,0,10,0,
%U A325165 13,12,12,8,0,0,0,0,0,0,11
%N A325165 Regular triangle read by rows where T(n,k) is the number of integer partitions of n whose inner lining partition has last (smallest) part equal to k.
%C A325165 The k-th part of the inner lining partition of an integer partition is the number of squares in its Young diagram that are k diagonal steps from the lower-right boundary. For example, the partition (6,5,5,3) has diagram
%C A325165 o o o o o o
%C A325165 o o o o o
%C A325165 o o o o o
%C A325165 o o o
%C A325165 which has diagonal distances from the lower-right boundary equal to
%C A325165 3 3 3 2 1 1
%C A325165 3 2 2 2 1
%C A325165 2 2 1 1 1
%C A325165 1 1 1
%C A325165 so the inner lining sequence is (9,6,4) with last part 4, so (6,5,5,3) is counted under T(19,4).
%H A325165 Andrew Howroyd, <a href="/A325165/b325165.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F A325165 G.f.: A(x,y) = 1 + Sum_{k>=1} x^(k^2)/((1 - y*x^k) * Product_{j=1..k-1} (1 - x^j))^2. - _Andrew Howroyd_, Jan 19 2023
%e A325165 Triangle begins:
%e A325165 1
%e A325165 0 1
%e A325165 0 0 2
%e A325165 0 0 0 3
%e A325165 0 1 0 0 4
%e A325165 0 2 0 0 0 5
%e A325165 0 3 2 0 0 0 6
%e A325165 0 4 4 0 0 0 0 7
%e A325165 0 5 6 3 0 0 0 0 8
%e A325165 0 7 8 6 0 0 0 0 0 9
%e A325165 0 9 10 9 4 0 0 0 0 0 10
%e A325165 0 13 12 12 8 0 0 0 0 0 0 11
%e A325165 0 17 16 15 12 5 0 0 0 0 0 0 12
%e A325165 0 24 20 18 16 10 0 0 0 0 0 0 0 13
%e A325165 0 31 28 21 20 15 6 0 0 0 0 0 0 0 14
%e A325165 0 42 36 27 24 20 12 0 0 0 0 0 0 0 0 15
%e A325165 0 54 50 33 28 25 18 7 0 0 0 0 0 0 0 0 16
%e A325165 0 71 64 45 32 30 24 14 0 0 0 0 0 0 0 0 0 17
%e A325165 0 90 86 57 40 35 30 21 8 0 0 0 0 0 0 0 0 0 18
%e A325165 Row n = 9 counts the following partitions (empty columns not shown):
%e A325165 (72) (63) (54) (9)
%e A325165 (333) (522) (432) (81)
%e A325165 (621) (531) (441) (711)
%e A325165 (5211) (4221) (3222) (6111)
%e A325165 (42111) (4311) (3321) (51111)
%e A325165 (321111) (32211) (22221) (411111)
%e A325165 (2211111) (33111) (3111111)
%e A325165 (222111) (21111111)
%e A325165 (111111111)
%t A325165 pml[ptn_]:=If[ptn=={},{},FixedPointList[If[#=={},{},DeleteCases[Rest[#]-1,0]]&,ptn][[-3]]];
%t A325165 Table[Length[Select[IntegerPartitions[n],Total[pml[#]]==k&]],{n,0,10},{k,0,n}]
%o A325165 (PARI) T(n) = {my(v=Vec(1+sum(k=1, sqrtint(n), x^(k^2)/((1-y*x^k)*prod(j=1, k-1, 1 - x^j + O(x^(n+1-k^2))))^2))); vector(#v, i, Vecrev(v[i], -i))}
%o A325165 { my(A=T(12)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 19 2023
%Y A325165 Row sums are A000041. Column k = 1 is A188674.
%Y A325165 Cf. A006918, A115720, A115994, A117485, A252464, A257990, A325163, A325166, A325168.
%K A325165 nonn,tabl
%O A325165 0,6
%A A325165 _Gus Wiseman_, Apr 05 2019