A182984 Total number of parts that are not the smallest part in all partitions of n.

0, 0, 0, 1, 2, 6, 9, 19, 29, 48, 73, 114, 161, 241, 340, 479, 662, 917, 1237, 1678, 2231, 2965, 3901, 5114, 6629, 8588, 11036, 14129, 17983, 22823, 28790, 36238, 45381, 56674, 70502, 87453, 108077, 133259, 163762, 200747, 245378, 299261

Offset

0,5

Comments

a(n) = sum of 2nd largest part in all partitions of n (if all parts are equal, then we assume that 0 is also a part). Example: a(5) = 6 because the sum of the 2nd largest parts in the partitions [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], and [1,1,1,1,1] is 0 + 1 + 2 + 1 + 1 + 1 + 0 = 6. - Emeric Deutsch, Dec 11 2015

Links

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

Formula

a(n) = A006128(n) - A092269(n), for n >= 1.

G.f.: g(x) = Sum(Sum(x^{q+i}/(1-x^q), q=i+1..infinity)/Product(1-x^q, q=i..infinity), i=1..infinity). - Emeric Deutsch, Nov 14 2015

a(n) = Sum(k*A264402(n,k), k>=1). - Emeric Deutsch, Dec 11 2015

Example

a(5) = 6 because the partitions of 5 are [5], [(4),1], [(3),2], [(3),1,1], [(2),(2),1], [(2),1,1,1] and [1,1,1,1,1], containing a total of 6 parts that are not the smallest part (shown between parentheses).

Maple

g := sum((sum(x^(q+i)/(1-x^q), q = i+1 .. 80))/(product(1-x^q, q = i .. 80)), i = 1 .. 80): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 47); # Emeric Deutsch, Nov 14 2015

Crossrefs

Cf. A006128, A092269, A116686, A264402.

Sequence in context: A361271 A082290 A325536 * A301798 A093840 A129233

Adjacent sequences: A182981 A182982 A182983 * A182985 A182986 A182987

Keyword

nonn

Author

Omar E. Pol, Jul 15 2011

Status

approved