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

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, 21, 26, 29, 30, 31, 39, 38, 42, 46, 49, 52, 61, 60, 68, 74, 77, 83, 94, 95, 104, 112, 122, 128, 143, 144, 159, 172, 181, 192, 212, 219, 237, 253, 271, 285

Offset: 1

Author

Clark Kimberling, Feb 16 2014

Keywords

• nonn
• easy

Examples

			a(8) = 3 counts these partitions:  53, 332, 11111111.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A118096, A237828.

Cf. A237757, A237825, A237826, A237827, A000041.

Programs

Mathematica

z = 64; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 3 Min[p] == Max[p]], {n, z}]     (* A237825*)
Table[Count[q[n], p_ /; 4 Min[p] == Max[p]], {n, z}]     (* A237826 *)
Table[Count[q[n], p_ /; 5 Min[p] == Max[p]], {n, z}]     (* A237827 *)
Table[Count[q[n], p_ /; 2 Min[p] + 1 == Max[p]], {n, z}] (* A237828 *)
Table[Count[q[n], p_ /; 2 Min[p] - 1 == Max[p]], {n, z}] (* A237829 *)
(* Second program: *)
kmax = 64;
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 *)
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 *)

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

Formula

G.f.: x + Sum_{k>=1} x^(3*k-1)/Product_{j=k..2*k-1} (1-x^j). - Seiichi Manyama, May 17 2023

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