A265250 Number of partitions of n having no parts strictly between the smallest and the largest part (n>=1).

1, 2, 3, 5, 7, 10, 13, 17, 20, 26, 29, 35, 39, 48, 48, 60, 61, 74, 73, 87, 86, 106, 99, 120, 112, 140, 130, 155, 143, 176, 159, 194, 180, 216, 186, 240, 209, 258, 234, 274, 243, 308, 261, 325, 289, 348, 297, 383, 314, 392, 356, 423, 355, 460, 372, 468, 422

Offset: 1

Author

Emeric Deutsch, Dec 25 2015

Keywords

• nonn
• look

Examples

			a(3) = 3 because we have [3], [1,2], [1,1,1] (all partitions of 3).
a(6) = 10 because we have all A000041(6) = 11 partitions of 6 except [1,2,3].
a(7) = 13 because we have all A000041(7) = 15 partitions of 7 except [1,2,4] and [1,1,2,3].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Jonathan Bloom, Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.

Crossrefs

Cf. A000041, A265249.

Cf. A000005, A002133, A116608, A309058.

Programs

Maple

g := add(x^i/(1-x^i), i = 1 .. 80)+add(add(x^(i+j)/((1-x^i)*(1-x^j)), j = i+1..80),i=1..80): gser := series(g,x=0,60): seq(coeff(gser,x,n),n=1..50);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
      `if`(t=1, `if`(irem(n, i)=0, 1, 0)+b(n, i-1, t),
       add(b(n-i*j, i-1, t-`if`(j=0, 0, 1)), j=0..n/i))))
    end:
a:= n-> b(n$2, 2):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 01 2016

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0, If[t == 1, If[Mod[n, i] == 0, 1, 0] + b[n, i - 1, t], Sum[b[n - i*j, i - 1, t - If[j == 0, 0, 1]], {j, 0, n/i}]]]]; a[n_] := b[n, n, 2]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

Formula

a(n) = A265249(n,0).

G.f.: G(x) = Sum_{i>=1} x^i/(1-x^i) + Sum_{i>=1} Sum_{j>=i+1} x^(i+j)/ ((1-x^i)*(1-x^j)).

a(n) = A116608(n,1) + A116608(n,2) = A000005(n) + A002133(n). - Seiichi Manyama, Sep 14 2023