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 A237828 #27 Jun 20 2025 07:12:43
%S A237828 0,0,0,1,1,2,4,4,6,9,10,12,17,18,22,27,31,34,42,45,53,61,66,72,86,92,
%T A237828 103,113,125,135,154,163,180,197,213,229,257,271,294,318,346,368,404,
%U A237828 426,463,497,532,564,616,651,700,747,798,844,912,962,1033,1097,1167,1231,1327,1397,1486,1576,1677
%N A237828 Number of partitions of n such that 2*(least part) + 1 = greatest part.
%C A237828 Also, the number of partitions p of n such that if h = max(p), then h is an (h,0)-separator of p; for example, a(10) counts these 9 partitions: 181, 271, 361, 262, 451, 352, 343, 23131, 1212121. - _Clark Kimberling_, Mar 24 2014
%H A237828 Vaclav Kotesovec, <a href="/A237828/b237828.txt">Table of n, a(n) for n = 1..10000</a>
%F A237828 G.f.: Sum_{k>=1} x^(3*k+1)/Product_{j=k..2*k+1} (1-x^j). - _Seiichi Manyama_, May 17 2023
%F A237828 a(n) ~ sqrt(phi) * exp(Pi*sqrt(2*n/15)) / (sqrt(2)* 5^(1/4) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 20 2025
%e A237828 a(8) = 4 counts these partitions: 3311, 3221, 32111, 311111.
%t A237828 z = 64; q[n_] := q[n] = IntegerPartitions[n];
%t A237828 Table[Count[q[n], p_ /; 3 Min[p] = = Max[p]], {n, z}] (* A237825*)
%t A237828 Table[Count[q[n], p_ /; 4 Min[p] = = Max[p]], {n, z}] (* A237826 *)
%t A237828 Table[Count[q[n], p_ /; 5 Min[p] = = Max[p]], {n, z}] (* A237827 *)
%t A237828 Table[Count[q[n], p_ /; 2 Min[p] + 1 = = Max[p]], {n, z}] (* A237828 *)
%t A237828 Table[Count[q[n], p_ /; 2 Min[p] - 1 = = Max[p]], {n, z}] (* A237829 *)
%t A237828 Table[Count[IntegerPartitions[n],_?(2*Min[#]+1==Max[#]&)],{n,60}] (* _Harvey P. Dale_, Jun 25 2017 *)
%t A237828 kmax = 65;
%t A237828 Sum[x^(3k+1)/Product[1-x^j, {j, k, 2k+1}], {k, 1, kmax}]/x + O[x]^kmax // CoefficientList[#, x]& (* _Jean-François Alcover_, May 30 2024, after _Seiichi Manyama_ *)
%t A237828 nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(2*k - 1))*(1 - x^(2*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax + 1)]; s += x^(3*k + 1)/(1 - x^k)/(1 - x^(2*k + 1))/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jun 18 2025 *)
%o A237828 (PARI) my(N=70, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(3*k+1)/prod(j=k, 2*k+1, 1-x^j)))) \\ _Seiichi Manyama_, May 17 2023
%Y A237828 Cf. A118096, A237829.
%Y A237828 Cf. A049820, A237757, A237825, A237826, A237827, A000041.
%K A237828 nonn,easy
%O A237828 1,6
%A A237828 _Clark Kimberling_, Feb 16 2014