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 A101707 #33 May 23 2021 02:52:32
%S A101707 0,0,1,0,2,1,4,2,7,6,13,11,22,22,38,39,63,69,103,114,165,189,262,301,
%T A101707 407,475,626,733,950,1119,1427,1681,2118,2503,3116,3678,4539,5360,
%U A101707 6559,7735,9400,11076,13372,15728,18886,22184,26501,31067,36947,43242,51210,59818,70576,82291,96750
%N A101707 Number of partitions of n having positive odd rank (the rank of a partition is the largest part minus the number of parts).
%C A101707 a(n) + A101708(n) = A064173(n).
%D A101707 George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.
%H A101707 Alois P. Heinz, <a href="/A101707/b101707.txt">Table of n, a(n) for n = 0..10000</a>
%H A101707 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a>
%F A101707 a(n) = (A000041(n) - A000025(n))/4. - _Vladeta Jovovic_, Dec 14 2004
%F A101707 G.f.: Sum((-1)^(k+1)*x^((3*k^2+k)/2)/(1+x^k), k=1..infinity)/Product(1-x^k, k=1..infinity). - _Vladeta Jovovic_, Dec 20 2004
%F A101707 a(n) = A340692(n)/2. - _Gus Wiseman_, Feb 07 2021
%e A101707 a(7)=2 because the only partitions of 7 with positive odd rank are 421 (rank=1) and 52 (rank=3).
%e A101707 From _Gus Wiseman_, Feb 07 2021: (Start)
%e A101707 Also the number of integer partitions of n into an even number of parts, the greatest of which is odd. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot) are:
%e A101707 11 . 31 32 33 52 53 54 55
%e A101707 1111 51 3211 71 72 73
%e A101707 3111 3221 3222 91
%e A101707 111111 3311 3321 3322
%e A101707 5111 5211 3331
%e A101707 311111 321111 5221
%e A101707 11111111 5311
%e A101707 7111
%e A101707 322111
%e A101707 331111
%e A101707 511111
%e A101707 31111111
%e A101707 1111111111
%e A101707 Also the number of integer partitions of n into an odd number of parts, the greatest of which is even. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot, A = 10) are:
%e A101707 2 . 4 221 6 421 8 432 A
%e A101707 211 222 22111 422 441 433
%e A101707 411 431 621 442
%e A101707 21111 611 22221 622
%e A101707 22211 42111 631
%e A101707 41111 2211111 811
%e A101707 2111111 22222
%e A101707 42211
%e A101707 43111
%e A101707 61111
%e A101707 2221111
%e A101707 4111111
%e A101707 211111111
%e A101707 (End)
%p A101707 b:= proc(n, i, r) option remember; `if`(n=0, max(0, r),
%p A101707 `if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
%p A101707 `if`(r<0, irem(i, 2), r))))
%p A101707 end:
%p A101707 a:= n-> b(n$2, -1)/2:
%p A101707 seq(a(n), n=0..55); # _Alois P. Heinz_, Jan 29 2021
%t A101707 Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&OddQ[Max[#]]&]],{n,0,30}] (* _Gus Wiseman_, Feb 10 2021 *)
%t A101707 b[n_, i_, r_] := b[n, i, r] = If[n == 0, Max[0, r],
%t A101707 If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 -
%t A101707 If[r < 0, Mod[i, 2], r]]]];
%t A101707 a[n_] := b[n, n, -1]/2;
%t A101707 a /@ Range[0, 55] (* _Jean-François Alcover_, May 23 2021, after _Alois P. Heinz_ *)
%Y A101707 Note: A-numbers of ranking sequences are in parentheses below.
%Y A101707 The even-rank version is A101708 (A340605).
%Y A101707 The even- but not necessarily positive-rank version is A340601 (A340602).
%Y A101707 The Heinz numbers of these partitions are (A340604).
%Y A101707 Allowing negative odd ranks gives A340692 (A340603).
%Y A101707 - Rank -
%Y A101707 A047993 counts balanced (rank zero) partitions (A106529).
%Y A101707 A064173 counts partitions of positive/negative rank (A340787/A340788).
%Y A101707 A064174 counts partitions of nonpositive/nonnegative rank (A324521/A324562).
%Y A101707 A101198 counts partitions of rank 1 (A325233).
%Y A101707 A257541 gives the rank of the partition with Heinz number n.
%Y A101707 - Odd -
%Y A101707 A000009 counts partitions into odd parts (A066208).
%Y A101707 A026804 counts partitions whose least part is odd.
%Y A101707 A027193 counts partitions of odd length/maximum (A026424/A244991).
%Y A101707 A058695 counts partitions of odd numbers (A300063).
%Y A101707 A339890 counts factorizations of odd length.
%Y A101707 A340385 counts partitions of odd length and maximum (A340386).
%Y A101707 Cf. A000041, A027187, A101709, A101199, A101200, A117409, A200750.
%K A101707 nonn
%O A101707 0,5
%A A101707 _Emeric Deutsch_, Dec 12 2004
%E A101707 More terms from _Joerg Arndt_, Oct 07 2012
%E A101707 a(0)=0 prepended by _Alois P. Heinz_, Jan 29 2021