A211870 -id:A211870 - cp's OEIS frontend

A116686 Total number of parts smaller than the largest part, in all partitions of n.

0, 0, 1, 3, 8, 15, 29, 48, 79, 123, 188, 276, 404, 575, 808, 1122, 1540, 2089, 2811, 3748, 4958, 6519, 8504, 11034, 14231, 18268, 23312, 29638, 37486, 47245, 59279, 74140, 92347, 114703, 141933, 175174, 215478, 264407, 323448, 394788, 480509, 583609

Offset: 1

Emeric Deutsch, Feb 23 2006

nonn

Also, sum over all partitions of n of the difference between the largest part and the smallest part. - Franklin T. Adams-Watters, Feb 29 2008

Row sums of A244966. - Omar E. Pol, Jul 19 2014

			a(5) = 8 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 8 parts that are smaller than the largest part (shown between parentheses).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A116685, A211870, A211881.

f:=sum(x^i*sum(x^j/(1-x^j),j=1..i-1)/product(1-x^q,q=1..i),i=2..55): fser:=series(f,x=0,50): seq(coeff(fser,x^n),n=1..47);

Table[Length[Flatten[Rest[Split[#]]&/@Select[IntegerPartitions[n], #[[1]]> #[[-1]]&]]],{n,50}] (* Harvey P. Dale, Jul 26 2016 *)

a(n) = Sum_{k>=0} k*A116685(n,k).
G.f.: Sum_{i>=1} (x^i*(Sum_{j=1..i-1} x^j/(1-x^j))/(Product_{q=1..i} (1-x^q))).
a(n) = A006128(n) - A046746(n). - Vladeta Jovovic, Feb 24 2006
a(n) = A211870(n) + A211881(n). - Alois P. Heinz, Feb 13 2013

A211881 Difference between sum of largest parts and sum of smallest parts of all partitions of n into an even number of parts.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 9, 16, 26, 41, 65, 95, 142, 202, 293, 403, 568, 766, 1054, 1399, 1886, 2469, 3276, 4237, 5538, 7094, 9162, 11628, 14856, 18704, 23670, 29590, 37130, 46109, 57428, 70885, 87685, 107634, 132324, 161595, 197545, 240091, 291990, 353302, 427624

Offset: 0

Alois P. Heinz, Feb 13 2013

nonn

			a(6) = 9: partitions of 6 into an even number of parts are [1,1,1,1,1,1], [2,2,1,1], [3,1,1,1], [3,3], [4,2], [5,1], difference between sum of largest parts and sum of smallest parts is (1+2+3+3+4+5) - (1+1+1+3+2+1) = 18 - 9 = 9.

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

Cf. A116686, A211870, A222045, A222048.

g:= proc(n, i) option remember; [`if`(n=i, n, 0), 0]+
      `if`(i>n, [0, 0], g(n, i+1)+(l-> [l[2], l[1]])(g(n-i, i)))
    end:
b:= proc(n, i) option remember;
      [`if`(n=i, n, 0), 0]+`if`(i<1, [0, 0], b(n, i-1)+
       `if`(n [l[2], l[1]])(b(n-i, i))))
    end:
a:= n-> g(n, 1)[2] -b(n, n)[2]:
seq(a(n), n=0..50);

g[n_, i_] := g[n, i] = {If[n==i, n, 0], 0} + If[i>n, {0, 0}, g[n, i+1] + Reverse[g[n-i, i]]]; b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i<1, {0, 0}, b[n, i-1] + If[nJean-François Alcover, Feb 15 2017, translated from Maple *)

a(n) = A222048(n) - A222045(n).

a(n) = A116686(n) - A211870(n).

cp's OEIS Frontend

A116686 Total number of parts smaller than the largest part, in all partitions of n.

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