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 A340601 #29 May 04 2024 05:09:29
%S A340601 1,1,0,3,1,5,3,11,8,18,16,34,33,57,59,98,105,159,179,262,297,414,478,
%T A340601 653,761,1008,1184,1544,1818,2327,2750,3480,4113,5137,6078,7527,8899,
%U A340601 10917,12897,15715,18538,22431,26430,31805,37403,44766,52556,62620,73379
%N A340601 Number of integer partitions of n of even rank.
%C A340601 The Dyson rank of a nonempty partition is its maximum part minus its number of parts. For this sequence, the rank of an empty partition is 0.
%H A340601 Alois P. Heinz, <a href="/A340601/b340601.txt">Table of n, a(n) for n = 0..10000</a>
%H A340601 Freeman J. Dyson, <a href="https://doi.org/10.1016/S0021-9800(69)80006-2">A new symmetry of partitions</a>, Journal of Combinatorial Theory 7.1 (1969): 56-61.
%H A340601 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a>
%F A340601 G.f.: 1 + Sum_{i, j>0} q^(i*j) * ( (1+(-1)^(i+j))/2 + Sum_{k>0} q^k * q_binomial(k,i-2) * (1+(-1)^(i+j+k))/2 ). - _John Tyler Rascoe_, Apr 15 2024
%F A340601 a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - _Vaclav Kotesovec_, Apr 17 2024
%e A340601 The a(1) = 1 through a(9) = 18 partitions (empty column indicated by dot):
%e A340601 (1) . (3) (22) (5) (42) (7) (44) (9)
%e A340601 (21) (41) (321) (43) (62) (63)
%e A340601 (111) (311) (2211) (61) (332) (81)
%e A340601 (2111) (322) (521) (333)
%e A340601 (11111) (331) (2222) (522)
%e A340601 (511) (4211) (531)
%e A340601 (2221) (32111) (711)
%e A340601 (4111) (221111) (4221)
%e A340601 (31111) (4311)
%e A340601 (211111) (6111)
%e A340601 (1111111) (32211)
%e A340601 (33111)
%e A340601 (51111)
%e A340601 (222111)
%e A340601 (411111)
%e A340601 (3111111)
%e A340601 (21111111)
%e A340601 (111111111)
%p A340601 b:= proc(n, i, r) option remember; `if`(n=0, 1-max(0, r),
%p A340601 `if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
%p A340601 `if`(r<0, irem(i, 2), r))))
%p A340601 end:
%p A340601 a:= n-> b(n$2, -1):
%p A340601 seq(a(n), n=0..55); # _Alois P. Heinz_, Jan 22 2021
%t A340601 Table[If[n==0,1,Length[Select[IntegerPartitions[n],EvenQ[Max[#]-Length[#]]&]]],{n,0,30}]
%t A340601 (* Second program: *)
%t A340601 b[n_, i_, r_] := b[n, i, r] = If[n == 0, 1 - Max[0, r], If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 - If[r < 0, Mod[i, 2], r]]]];
%t A340601 a[n_] := b[n, n, -1];
%t A340601 a /@ Range[0, 55] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)
%o A340601 (PARI)
%o A340601 p_q(k) = {prod(j=1, k, 1-q^j); }
%o A340601 GB_q(N, M)= {if(N>=0 && M>=0, p_q(N+M)/(p_q(M)*p_q(N)), 0 ); }
%o A340601 A_q(N) = {my(q='q+O('q^N), g=1+sum(i=1,N, sum(j=1,N/i, q^(i*j) * ( ((1/2)*(1+(-1)^(i+j))) + sum(k=1,N-(i*j), ((q^k)*GB_q(k,i-2)) * ((1/2)*(1+(-1)^(i+j+k)))))))); Vec(g)}
%o A340601 A_q(50) \\ _John Tyler Rascoe_, Apr 15 2024
%Y A340601 Note: Heinz numbers are given in parentheses below.
%Y A340601 The positive case is A101708 (A340605).
%Y A340601 The Heinz numbers of these partitions are A340602.
%Y A340601 The odd version is A340692 (A340603).
%Y A340601 - Rank -
%Y A340601 A047993 counts partitions of rank 0 (A106529).
%Y A340601 A072233 counts partitions by sum and length.
%Y A340601 A101198 counts partitions of rank 1 (A325233).
%Y A340601 A101707 counts partitions of odd positive rank (A340604).
%Y A340601 A101708 counts partitions of even positive rank (A340605).
%Y A340601 A257541 gives the rank of the partition with Heinz number n.
%Y A340601 A340653 counts factorizations of rank 0.
%Y A340601 - Even -
%Y A340601 A024430 counts set partitions of even length.
%Y A340601 A027187 counts partitions of even length (A028260).
%Y A340601 A027187 (also) counts partitions of even maximum (A244990).
%Y A340601 A034008 counts compositions of even length.
%Y A340601 A035363 counts partitions into even parts (A066207).
%Y A340601 A052841 counts ordered set partitions of even length.
%Y A340601 A058696 counts partitions of even numbers (A300061).
%Y A340601 A067661 counts strict partitions of even length (A030229).
%Y A340601 A236913 counts even-length partitions of even numbers (A340784).
%Y A340601 A339846 counts factorizations of even length.
%Y A340601 Cf. A000041, A006141, A039900, A064174, A067538, A117409, A200750, A324516.
%K A340601 nonn
%O A340601 0,4
%A A340601 _Gus Wiseman_, Jan 21 2021