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 A064174 #80 May 26 2023 08:54:24
%S A064174 1,1,2,3,4,6,9,12,17,23,31,42,56,73,96,125,161,207,265,336,426,536,
%T A064174 672,840,1046,1296,1603,1975,2425,2970,3628,4417,5367,6503,7861,9482,
%U A064174 11412,13702,16423,19642,23447,27938,33231,39453,46767,55342,65386,77135
%N A064174 Number of partitions of n with nonnegative rank.
%C A064174 The rank of a partition is the largest summand minus the number of summands.
%C A064174 This sequence (up to proof) equals "partitions of 2n with even number of parts, ending in 1, with max descent of 1, where the number of odd parts in odd places equals the number of odd parts in even places. (See link and 2nd Mathematica line.) - _Wouter Meeussen_, Mar 29 2013
%C A064174 Number of partitions p of n such that max(max(p), number of parts of p) is a part of p. - _Clark Kimberling_, Feb 28 2014
%C A064174 From _Gus Wiseman_, Mar 09 2019: (Start)
%C A064174 Also the number of integer partitions of n with maximum part greater than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324521. For example, the a(1) = 1 through a(8) = 12 partitions are:
%C A064174 (1) (2) (3) (4) (5) (6) (7) (8)
%C A064174 (21) (22) (32) (33) (43) (44)
%C A064174 (31) (41) (42) (52) (53)
%C A064174 (311) (51) (61) (62)
%C A064174 (321) (322) (71)
%C A064174 (411) (331) (332)
%C A064174 (421) (422)
%C A064174 (511) (431)
%C A064174 (4111) (521)
%C A064174 (611)
%C A064174 (4211)
%C A064174 (5111)
%C A064174 Also the number of integer partitions of n with maximum part less than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324562. For example, the a(1) = 1 through a(8) = 12 partitions are:
%C A064174 (1) (11) (21) (22) (221) (222) (322) (332)
%C A064174 (111) (211) (311) (321) (331) (2222)
%C A064174 (1111) (2111) (2211) (2221) (3221)
%C A064174 (11111) (3111) (3211) (3311)
%C A064174 (21111) (4111) (4211)
%C A064174 (111111) (22111) (22211)
%C A064174 (31111) (32111)
%C A064174 (211111) (41111)
%C A064174 (1111111) (221111)
%C A064174 (311111)
%C A064174 (2111111)
%C A064174 (11111111)
%C A064174 (End)
%H A064174 Seiichi Manyama, <a href="/A064174/b064174.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%H A064174 Cristina Ballantine and Mircea Merca, <a href="https://arxiv.org/abs/1710.05960">Bisected theta series, least r-gaps in partitions, and polygonal numbers</a>, arXiv:1710.05960 [math.CO], 2017.
%H A064174 Rekha Biswal, <a href="http://mathoverflow.net/questions/125709">bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n</a>, Mathoverflow.
%H A064174 Mircea Merca, <a href="https://arxiv.org/abs/2006.07705">Rank partition functions and truncated theta identities</a>, arXiv:2006.07705 [math.CO], 2020.
%F A064174 a(n) = (A000041(n) + A047993(n))/2.
%F A064174 a(n) = p(n-1) - p(n-5) + p(n-12) - ... -(-1)^k*p(n-(3*k^2-k)/2) + ..., where p() is A000041(). - _Vladeta Jovovic_, Aug 04 2004
%F A064174 G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1 - x^(n+k-1))/(1 - x^k). - _Paul D. Hanna_, Aug 03 2015
%F A064174 A064173(n) + a(n) = A000041(n). - _R. J. Mathar_, Feb 22 2023
%F A064174 G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2). - _Seiichi Manyama_, May 21 2023
%e A064174 a(20) = p(19) - p(15) + p(8) = 490 - 176 + 22 = 336.
%p A064174 f:= n -> add((-1)^(k+1)*combinat:-numbpart(n-(3*k^2-k)/2),k=1..floor((1+sqrt(24*n+1))/6)):
%p A064174 map(f, [$1..100]); # _Robert Israel_, Aug 03 2015
%t A064174 Table[Count[IntegerPartitions[n], q_ /; First[q] >= Length[q]], {n, 16}]
%t A064174 (* also *)
%t A064174 Table[Count[IntegerPartitions[2n],q_/;Last[q]===1 && Max[q-PadRight[Rest[q],Length[q]]]<=1 && Count[First/@Partition[q,2],_?OddQ]==Count[Last/@Partition[q,2],_?OddQ]],{n,16}]
%t A064174 (* also *)
%t A064174 Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[Max[p], Length[p]]]], {n, 50}] (* _Clark Kimberling_, Feb 28 2014 *)
%o A064174 (PARI) {a(n) = my(A=1); A = sum(m=0,n,x^m*prod(k=1,m,(1-x^(m+k-1))/(1-x^k +x*O(x^n)))); polcoeff(A,n)}
%o A064174 for(n=1,60,print1(a(n),", ")) \\ _Paul D. Hanna_, Aug 03 2015
%o A064174 (PARI) my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2))) \\ _Seiichi Manyama_, May 21 2023
%Y A064174 Cf. A063995, A064173 (complement).
%Y A064174 Row sums of triangle A105806.
%Y A064174 Cf. A003114, A006141, A039900, A047993, A090858, A106529, A133121.
%Y A064174 Cf. A324516, A324518, A324520, A324521, A324522, A324560, A324562, A324572.
%K A064174 nonn
%O A064174 1,3
%A A064174 _Vladeta Jovovic_, Sep 20 2001
%E A064174 Mathematica programs modified by _Clark Kimberling_, Feb 12 2014