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A105039 Number of compositions of n with unique smallest part. 3
1, 1, 3, 3, 8, 11, 20, 34, 59, 96, 167, 282, 475, 800, 1352, 2275, 3828, 6426, 10785, 18085, 30297, 50698, 84770, 141623, 236425, 394381, 657380, 1094975, 1822628, 3031843, 5040129, 8373594, 13903588, 23072567, 38267330, 63435438, 105103059 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

FORMULA

G.f.: Sum(k*x^(2*k-1)/((1-x^k)*(1-x)^(k-1)), k=1..infinity).

Also (1-x)^2*Sum(x^k/(1-x-x^(k+1))^2, k=1..infinity). - Vladeta Jovovic, Apr 05 2005

a(n) = 1 + sum(k=2..[(n+3)/2], k * sum(s=1..[(n-1)/k], binomial(n-k*s-1, k-2) ) ). - Max Alekseyev, Apr 15 2005

a(n) ~ (2*sqrt(5)-4)/10 * n * ((1+sqrt(5))/2)^n. - Vaclav Kotesovec, May 02 2014

EXAMPLE

a(5) = 8 because we have 5, 14, 41, 23, 32, 122, 212 and 221.

MAPLE

G:= sum(k*x^(2*k-1)/((1-x^k)*(1-x)^(k-1)), k=1..70): Gser:=series(G, x=0, 44): seq(coeff(Gser, x^n), n=1..41); # Emeric Deutsch, Apr 13 2005

MATHEMATICA

nn=37; Drop[CoefficientList[Series[Sum[x^j/(1-x^(j+1)/(1-x))^2, {j, 1, nn}], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Mar 31 2014 *)

PROG

(PARI) a(n)=1+sum(k=2, (n+3)\2, k*sum(s=1, (n-1)\k, binomial(n-k*s-1, k-2))) (Alekseyev)

CROSSREFS

Cf. A079501, A097979.

Column k=1 of A238342.

Sequence in context: A233174 A185350 A279910 * A276552 A213030 A303902

Adjacent sequences:  A105036 A105037 A105038 * A105040 A105041 A105042

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Apr 03 2005

EXTENSIONS

More terms from Emeric Deutsch and Max Alekseyev, Apr 13 2005

STATUS

approved

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Last modified July 12 21:31 EDT 2020. Contains 335669 sequences. (Running on oeis4.)