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

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

0,8

Links

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

Formula

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

Example

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.

Maple

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

Crossrefs

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

Sequence in context: A322072 A304627 A168193 * A116658 A210065 A208850

Adjacent sequences: A182974 A182975 A182976 * A182978 A182979 A182980

Keyword

nonn

Author

Omar E. Pol, Jul 17 2011

Extensions

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

Status

approved