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

0,5

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Formula

G.f.: Sum_{k>=0} x^(k^2)/(1-x)^k.

Example

a(7)=5 because we have 7, 4+3, 3+4, 5+2 and 2+5.

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

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 *)

Prog

(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

Crossrefs

Cf. A003114, A077229.

Sequence in context: A218930 A350391 A026483 * A017899 A003520 A101915

Adjacent sequences: A098128 A098129 A098130 * A098132 A098133 A098134

Keyword

easy,nonn

Author

Vladeta Jovovic, Sep 27 2004

Extensions

More terms from Emeric Deutsch, Mar 29 2005

Prepended a(0)=1 to match g.f., Joerg Arndt, Apr 22 2014

Status

approved