A265249 -id:A265249 - cp's OEIS frontend

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

Emeric Deutsch, Dec 25 2015

			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].

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.

Cf. A000041, A265249.

Cf. A000005, A002133, A116608, A309058.

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

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 *)

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

A182977 Total number of parts that are neither the smallest part nor the largest part in all partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 6, 12, 22, 39, 66, 103, 159, 243, 352, 510, 721, 1011, 1391, 1903, 2557, 3436, 4549, 5999, 7824, 10187, 13132, 16886, 21544, 27414, 34657, 43703, 54797, 68558, 85328, 105963, 131028, 161664, 198710

Offset: 0

Omar E. Pol, Jul 17 2011

nonn

			For n = 6 the partitions of 6 are
6
5 + 1
4 + 2
4 + 1 + 1
3 + 3
3 + (2) + 1 .......... the "2" is the part that counts.
3 + 1 + 1 + 1
2 + 2 + 2
2 + 2 + 1 + 1
2 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1
There is only one part which is neither the smallest part nor the largest part in all partitions of 6, so a(6) = 1.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A006128, A046746, A092269, A116686, A182978, A182984, A265249.

g := add(add((add(x^(i+j+k)/(1-x^k), k = i+1 .. j-1))/(mul(1-x^k, k = i .. j)), j = i+1 .. 80), i = 1 .. 80): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 45); # Emeric Deutsch, Dec 25 2015

a(n) = A006128(n) - A182978(n).
G.f.: g(x) = Sum_{i>=1} Sum_{j>=i+1} (Sum_{k=i+1..j-1} x^{i+j+k}/(1-x^k)/Product_{k=i..j}(1-x^k)). - Emeric Deutsch, Dec 25 2015
a(n) = Sum_{k>=0} k*A265249(n,k). - Emeric Deutsch, Dec 25 2015

a(12) corrected and more terms a(13)-a(40) from David Scambler, Jul 18 2011

cp's OEIS Frontend

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

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