A098131 Number of compositions of n where the smallest part is greater than or equal to the number of parts.
1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 33, 41, 51, 64, 81, 103, 131, 166, 209, 261, 323, 397, 486, 594, 726, 888, 1087, 1331, 1629, 1991, 2428, 2952, 3577, 4320, 5202, 6249, 7493, 8973, 10736, 12838, 15345, 18334, 21894, 26127, 31149, 37092, 44107, 52368
Offset: 0
Examples
a(7)=5 because we have 7, 4+3, 3+4, 5+2 and 2+5.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Programs
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Maple
G:=sum(x^(k^2)/(1-x)^k, k=0..20): Gser:=series(G,x=0,56): seq( coeff( Gser,x^n), n=0..54); # Emeric Deutsch
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Mathematica
nmax = 60; Flatten[{1, Rest[CoefficientList[Series[Sum[x^(k^2)/(1-x)^k, {k, 1, Sqrt[nmax]}], {x, 0, nmax}], x]]}] (* Vaclav Kotesovec, Nov 11 2018 *) -
PARI
my(N=66,x='x+O('x^N)); Vec(sum(n=1,N,x^(n*n)*(1)/(1-x)^n)) \\ Joerg Arndt, Jan 23 2024
Formula
G.f.: Sum_{k>=0} x^(k^2)/(1-x)^k.
Extensions
More terms from Emeric Deutsch, Mar 29 2005
Prepended a(0)=1 to match g.f., Joerg Arndt, Apr 22 2014