0,5

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

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

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

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).

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

Cf. A006128, A092269, A116686, A264402.

Sequence in context: A361271 A082290 A325536 * A301798 A093840 A129233

Adjacent sequences: A182981 A182982 A182983 * A182985 A182986 A182987

Omar E. Pol, Jul 15 2011

approved