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A015740 Number of 5's in all the partitions of n into distinct parts. 2
0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 4, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 45, 53, 62, 72, 84, 97, 112, 130, 150, 172, 199, 228, 260, 298, 340, 386, 440, 500, 566, 642, 727, 820, 926, 1044, 1174, 1321, 1484, 1664, 1866, 2090 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

FORMULA

G.f.: x^5*product(j>=1, 1+x^j )/(1+x^5). - Emeric Deutsch, Apr 17 2006

Corresponding g.f. for "number of k's" is x^k/(1+x^k)*prod(n>=1, 1+x^n ). [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)=2 because in the 8 (=A000009(9)) 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] we have altogether two parts equal to 5.

MAPLE

g:=x^5*product(1+x^j, j=1..60)/(1+x^5): gser:=series(g, x=0, 57): seq(coeff(gser, x, n), n=1..54); # Emeric Deutsch, Apr 17 2006

MATHEMATICA

nmax = 100; Rest[CoefficientList[Series[x^5/(1+x^5) * Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 30 2015 *)

Table[Count[Flatten@Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], 5], {n, 54}] (* Robert Price, May 16 2020 *)

CROSSREFS

Sequence in context: A274759 A274157 A005863 * A015750 A275442 A303846

Adjacent sequences:  A015737 A015738 A015739 * A015741 A015742 A015743

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified June 12 19:24 EDT 2021. Contains 344959 sequences. (Running on oeis4.)