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 A339737 #17 Jan 14 2024 11:50:22
%S A339737 1,1,0,1,1,0,2,0,1,0,2,1,1,1,0,3,1,1,1,1,0,4,1,2,2,1,1,0,5,1,3,2,2,1,
%T A339737 1,0,6,2,3,4,3,2,1,1,0,8,2,4,5,4,3,2,1,1,0,10,2,5,7,6,5,3,2,1,1,0,12,
%U A339737 3,6,8,9,6,5,3,2,1,1,0,15,3,8,11,11,10,7,5,3,2,1,1,0
%N A339737 Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.
%C A339737 We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.
%H A339737 Andrew Howroyd, <a href="/A339737/b339737.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A339737 George E. Andrews and David Newman, <a href="https://doi.org/10.1007/s00026-019-00427-w">Partitions and the Minimal Excludant</a>, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
%H A339737 Brian Hopkins, James A. Sellers, and Dennis Stanton, <a href="https://arxiv.org/abs/2009.10873">Dyson's Crank and the Mex of Integer Partitions</a>, arXiv:2009.10873 [math.CO], 2020.
%H A339737 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mex_(mathematics)">Mex (mathematics)</a>
%e A339737 Triangle begins:
%e A339737 1
%e A339737 1 0
%e A339737 1 1 0
%e A339737 2 0 1 0
%e A339737 2 1 1 1 0
%e A339737 3 1 1 1 1 0
%e A339737 4 1 2 2 1 1 0
%e A339737 5 1 3 2 2 1 1 0
%e A339737 6 2 3 4 3 2 1 1 0
%e A339737 8 2 4 5 4 3 2 1 1 0
%e A339737 10 2 5 7 6 5 3 2 1 1 0
%e A339737 12 3 6 8 9 6 5 3 2 1 1 0
%e A339737 15 3 8 11 11 10 7 5 3 2 1 1 0
%e A339737 18 4 9 13 15 13 10 7 5 3 2 1 1 0
%e A339737 22 5 10 17 19 18 14 11 7 5 3 2 1 1 0
%e A339737 27 5 13 20 24 23 20 14 11 7 5 3 2 1 1 0
%e A339737 For example, row n = 9 counts the following partitions:
%e A339737 (3321) (432) (333) (54) (522) (63) (72) (81) (9)
%e A339737 (22221) (3222) (4311) (441) (531) (621) (711)
%e A339737 (32211) (33111) (4221) (5211) (6111)
%e A339737 (222111) (3111111) (42111) (51111)
%e A339737 (321111) (411111)
%e A339737 (2211111)
%e A339737 (21111111)
%e A339737 (111111111)
%t A339737 maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
%t A339737 Table[Length[Select[IntegerPartitions[n],maxgap[#]==k&]],{n,0,15},{k,0,n}]
%o A339737 (PARI)
%o A339737 S(n,k)={if(k>n, O(x*x^n), x^k*(S(n-k,k+1) + 1)/(1 - x^k))}
%o A339737 ColGf(k,n) = {(k==0) + S(n,k+1)/prod(j=1, k-1, 1 - x^j + O(x^max(1,n-k)))}
%o A339737 A(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
%o A339737 { my(M=A(10)); for(i=1, #M, print(M[i,1..i])) } \\ _Andrew Howroyd_, Jan 13 2024
%Y A339737 Column k = 0 is A000009.
%Y A339737 Row sums are A000041.
%Y A339737 Central diagonal is A000041.
%Y A339737 Column k = 1 is A087897.
%Y A339737 The version for least gap is A264401, with Heinz number encoding A257993.
%Y A339737 The version for greatest difference is A286469 or A286470.
%Y A339737 An encoding (of greatest gap) using Heinz numbers is A339662.
%Y A339737 A000070 counts partitions with a selected part.
%Y A339737 A006128 counts partitions with a selected position.
%Y A339737 A015723 counts strict partitions with a selected part.
%Y A339737 A048004 counts compositions by greatest part.
%Y A339737 A056239 adds up prime indices, row sums of A112798.
%Y A339737 A064391 is the version for crank.
%Y A339737 A064428 counts partitions of nonnegative crank.
%Y A339737 A073491 list numbers with gap-free prime indices.
%Y A339737 A107428 counts gap-free compositions.
%Y A339737 A238709/A238710 counts partitions by least/greatest difference.
%Y A339737 A342050/A342051 have prime indices with odd/even least gap.
%Y A339737 Cf. A001223, A002110, A018818, A063250, A088860, A098743, A279945.
%K A339737 nonn,tabl
%O A339737 0,7
%A A339737 _Gus Wiseman_, Apr 20 2021
%E A339737 Offset corrected by _Andrew Howroyd_, Jan 13 2024