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A237829 Number of partitions of n such that 2*(least part) - 1 = greatest part.

%I A237829 #23 Jun 20 2025 07:08:32
%S A237829 1,1,1,1,2,1,2,3,2,2,5,3,4,5,5,6,8,6,8,10,10,10,15,12,14,17,18,20,23,
%T A237829 21,26,29,30,31,39,38,42,46,49,52,61,60,68,74,77,83,94,95,104,112,122,
%U A237829 128,143,144,159,172,181,192,212,219,237,253,271,285
%N A237829 Number of partitions of n such that 2*(least part) - 1 = greatest part.
%H A237829 Vaclav Kotesovec, <a href="/A237829/b237829.txt">Table of n, a(n) for n = 1..10000</a>
%F A237829 G.f.: x + Sum_{k>=1} x^(3*k-1)/Product_{j=k..2*k-1} (1-x^j). - _Seiichi Manyama_, May 17 2023
%F A237829 a(n) ~ exp(Pi*sqrt(2*n/15)) / (sqrt(2)* 5^(1/4) * phi^(3/2) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 20 2025
%e A237829 a(8) = 3 counts these partitions:  53, 332, 11111111.
%t A237829 z = 64; q[n_] := q[n] = IntegerPartitions[n];
%t A237829 Table[Count[q[n], p_ /; 3 Min[p] == Max[p]], {n, z}]     (* A237825*)
%t A237829 Table[Count[q[n], p_ /; 4 Min[p] == Max[p]], {n, z}]     (* A237826 *)
%t A237829 Table[Count[q[n], p_ /; 5 Min[p] == Max[p]], {n, z}]     (* A237827 *)
%t A237829 Table[Count[q[n], p_ /; 2 Min[p] + 1 == Max[p]], {n, z}] (* A237828 *)
%t A237829 Table[Count[q[n], p_ /; 2 Min[p] - 1 == Max[p]], {n, z}] (* A237829 *)
%t A237829 (* Second program: *)
%t A237829 kmax = 64;
%t A237829 Sum[x^(3k-1)/Product[1-x^j, {j, k, 2k-1}], {k, 1, kmax}]/x+1+O[x]^kmax // CoefficientList[#, x]& (* _Jean-François Alcover_, May 30 2024, after _Seiichi Manyama_ *)
%t A237829 nmax = 100; p = 1; s = x; 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))/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jun 18 2025 *)
%o A237829 (PARI) my(N=70, x='x+O('x^N)); Vec(x+sum(k=1, N, x^(3*k-1)/prod(j=k, 2*k-1, 1-x^j))) \\ _Seiichi Manyama_, May 17 2023
%Y A237829 Cf. A118096, A237828.
%Y A237829 Cf. A237757, A237825, A237826, A237827, A000041.
%K A237829 nonn,easy
%O A237829 1,5
%A A237829 _Clark Kimberling_, Feb 16 2014