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A015737
Number of 3's in partitions of n into distinct parts.
2
0, 0, 1, 1, 1, 1, 1, 2, 3, 4, 4, 5, 6, 8, 10, 12, 14, 17, 20, 24, 29, 34, 40, 47, 55, 64, 75, 87, 101, 117, 135, 155, 179, 205, 235, 269, 307, 350, 399, 453, 514, 583, 660, 746, 843, 950, 1070, 1205, 1354, 1520, 1705, 1910, 2138, 2392
OFFSET
1,8
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (first 70 terms from Vincenzo Librandi)
FORMULA
G.f.: (x^3/(1 + x^3)) * Product_{j >= 1} (1 + x^j). - Emeric Deutsch, Apr 17 2006
Corresponding g.f. for "number of k's" is (x^k/(1 + x^k)) * Product_{j >= 1} (1 + x^j). - Joerg Arndt, Feb 20 2014
a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015
EXAMPLE
a(9) = 3 because in the eight partitions of 9 into distinct parts, namely [9], [8, 1], [7, 2], [6, 3], [6, 2, 1], [5, 4], [5, 3, 1] and [4, 3, 2], only three contain 3.
MAPLE
g:=x^3*product(1+x^j, j=1..60)/(1+x^3): gser:=series(g, x=0, 57): seq(coeff(gser, x, n), n=1..54); # Emeric Deutsch, Apr 17 2006
MATHEMATICA
Table[Count[Select[IntegerPartitions[n], Length[Union[#]] == Length[#] &], _?(MemberQ[#, 3] &)], {n, 60}] (* Harvey P. Dale, Aug 19 2011 *)
nmax = 100; Rest[CoefficientList[Series[x^3/(1 + x^3) * Product[1 + x^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 30 2015 *)
CROSSREFS
Cf. A000009.
Sequence in context: A065328 A049877 A029063 * A015745 A375476 A017854
KEYWORD
nonn
EXTENSIONS
Example clarified by Harvey P. Dale, Aug 19 2011
STATUS
approved