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 A237753 #30 Oct 18 2024 03:58:31
%S A237753 0,1,0,0,1,1,1,2,1,2,3,4,5,7,7,9,12,15,17,23,27,34,42,50,60,75,87,106,
%T A237753 128,154,182,222,260,311,369,437,515,613,716,845,993,1166,1361,1599,
%U A237753 1861,2176,2534,2950,3422,3983,4605,5339,6174,7136,8227,9500,10928
%N A237753 Number of partitions of n such that 2*(greatest part) = (number of parts).
%C A237753 Also, the number of partitions of n such that (greatest part) = 2*(number of parts); hence, the number of partitions of n such that (rank + greatest part) = 0.
%H A237753 Vaclav Kotesovec, <a href="/A237753/b237753.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Seiichi Manyama)
%H A237753 Vaclav Kotesovec, <a href="/A237753/a237753.jpg">Graph - the asymptotic ratio (20000 terms)</a>
%F A237753 G.f.: Sum_{k>=1} x^(3*k-1) * Product_{j=1..k-1} (1-x^(2*k+j-1))/(1-x^j). - _Seiichi Manyama_, Jan 24 2022
%F A237753 a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (4 * 3^(3/2) * n^2). - _Vaclav Kotesovec_, Oct 17 2024
%e A237753 a(8) = 2 counts these partitions: 311111, 2222.
%t A237753 z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] = = Length[p]], {n, z}]
%t A237753 (* or *)
%t A237753 nmax = 100; Rest[CoefficientList[Series[Sum[x^(3*k-1) * Product[(1 - x^(2*k+j-1)) / (1 - x^j), {j, 1, k-1}], {k, 1, nmax/3 + 1}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Oct 15 2024 *)
%t A237753 nmax = 100; p = x; s = x; Do[p = Normal[Series[p*x^3*(1 - x^(3*k - 1))*(1 - x^(3*k))*(1 - x^(3*k + 1))/((1 - x^(2*k + 1))*(1 - x^(2*k))*(1 - x^k)), {x, 0, nmax}]]; s += p;, {k, 1, nmax/3 + 1}]; Take[CoefficientList[s, x], nmax] (* _Vaclav Kotesovec_, Oct 16 2024 *)
%o A237753 (PARI) my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(3*k-1)*prod(j=1, k-1, (1-x^(2*k+j-1))/(1-x^j))))) \\ _Seiichi Manyama_, Jan 24 2022
%Y A237753 Column 2 of A350879.
%Y A237753 Cf. A064173, A237751, A237752, A237754-A237757, A350893.
%K A237753 nonn,easy
%O A237753 1,8
%A A237753 _Clark Kimberling_, Feb 13 2014