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A116686 Total number of parts smaller than the largest part, in all partitions of n. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
a(n) = Sum(k*A116685(n,k), k>=0).
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
a(n) = A211870(n) + A211881(n). - Alois P. Heinz, Feb 13 2013
Row sums of A244966. - Omar E. Pol, Jul 19 2014
LINKS
FORMULA
G.f.: sum[x^i*sum(x^j/(1-x^j),j=1..i-1)/product(1-x^q, q=1..i)], i=1..infinity).
a(n) = A006128(n) - A046746(n). - Vladeta Jovovic, Feb 24 2006
EXAMPLE
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).
MAPLE
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);
MATHEMATICA
Table[Length[Flatten[Rest[Split[#]]&/@Select[IntegerPartitions[n], #[[1]]> #[[-1]]&]]], {n, 50}] (* Harvey P. Dale, Jul 26 2016 *)
CROSSREFS
Sequence in context: A294426 A097589 A015631 * A317252 A135350 A068038
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 23 2006
STATUS
approved

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Last modified July 23 20:45 EDT 2024. Contains 374559 sequences. (Running on oeis4.)