login
Total number of parts that are not the smallest part in all partitions of n.
6

%I #30 Dec 14 2015 03:50:14

%S 0,0,0,1,2,6,9,19,29,48,73,114,161,241,340,479,662,917,1237,1678,2231,

%T 2965,3901,5114,6629,8588,11036,14129,17983,22823,28790,36238,45381,

%U 56674,70502,87453,108077,133259,163762,200747,245378,299261

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

%C 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

%H Alois P. Heinz, <a href="/A182984/b182984.txt">Table of n, a(n) for n = 0..1000</a>

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

%F 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

%F a(n) = Sum(k*A264402(n,k), k>=1). - _Emeric Deutsch_, Dec 11 2015

%e 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).

%p 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

%Y Cf. A006128, A092269, A116686, A264402.

%K nonn

%O 0,5

%A _Omar E. Pol_, Jul 15 2011