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A118084 Number of partitions of n such that largest part k occurs at most floor(k/2) times. 5
0, 1, 2, 3, 5, 7, 11, 16, 23, 33, 46, 63, 86, 116, 153, 203, 265, 345, 444, 571, 727, 925, 1166, 1468, 1836, 2293, 2845, 3525, 4345, 5347, 6550, 8011, 9758, 11867, 14380, 17399, 20984, 25269, 30341, 36376, 43500, 51943, 61877, 73608, 87373, 103571 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Also number of partitions of n such that if the number of parts is k, then the smallest part is at most floor(k/2). Example: a(8)=16 because we have [7,1],[6,1,1],[5,2,1],[4,3,1],[5,1,1,1],[4,2,1,1],[3,3,1,1],[3,2,2,1],[2,2,2,2],[4,1,1,1,1],[3,2,1,1,1],[2,2,2,1,1],[3,1,1,1,1,1],[2,2,1,1,1,1],[2,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1].
LINKS
FORMULA
G.f.=sum(x^k*(1-x^(k(floor(k/2))))/product(1-x^j, j=1..k), k=1..infinity).
EXAMPLE
a(8)=16 because we have [8],[7,1],[6,2],[6,1,1],[5,3],[5,2,1],[5,1,1,1],[4,4],[4,3,1],[4,2,2],[4,2,1,1],[4,1,1,1,1],[3,2,2,1],[3,2,1,1,1],[3,1,1,1,1,1] and [2,1,1,1,1,1,1].
MAPLE
g:=sum(x^k*(1-x^(k*(floor(k/2))))/product(1-x^j, j=1..k), k=1..85): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=1..50);
MATHEMATICA
z=55 ; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := Length[p];
Table[Count[q[n], p_ /; 2 Min[p] <= t[p]], {n, z}] (* Clark Kimberling, Feb 15 2014 *)
CROSSREFS
Sequence in context: A024791 A373298 A178240 * A232481 A232482 A332062
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 12 2006
STATUS
approved

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Last modified June 18 21:09 EDT 2024. Contains 373487 sequences. (Running on oeis4.)