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

Author

Omar E. Pol, Jul 15 2011

Keywords

• nonn

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

Examples

			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).
		

Links

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

Crossrefs

Cf. A006128, A092269, A116686, A264402.

Programs

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

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