A077229 Number of compositions of n where the largest part is less than or equal to the number of parts.

1, 1, 1, 3, 5, 11, 23, 48, 98, 204, 421, 863, 1766, 3606, 7341, 14913, 30233, 61175, 123589, 249344, 502443, 1011366, 2033894, 4086975, 8206833, 16469875, 33035611, 66234372, 132745859, 265961487, 532717894, 1066778687, 2135822457, 4275459730, 8557335141, 17125445575, 34268965676, 68568213419, 137187103849, 274458924246

Offset: 0

Author

Henry Bottomley, Oct 29 2002

Keywords

• nonn

Examples

			a(5)=11 since 5 can be written as 1+1+1+1+1, 1+1+1+2, 1+1+2+1, 1+1+3, 1+2+1+1, 1+2+2, 1+3+1, 2+1+1+1, 2+1+2, 2+2+1, or 3+1+1; but not as 2+3 since then the largest part (3) would be greater than the number of parts (2).
		

Links

• Index entries for sequences related to compositions

Crossrefs

Row sums of A077227.

Cf. A064174, A348125.

Programs

Mathematica

Table[SeriesCoefficient[1 + Sum[x^k*((1-x^k)/(1-x))^k,{k,1,n}],{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, May 01 2014 *)

Formula

G.f.: 1 + Sum_{k>=0} ((x^(k+1)-x)/(x-1))^k. - Vladeta Jovovic, Sep 24 2004

G.f.: 1 + Sum_{n>=1} q^n * ( (1-q^n)/(1-q) )^n, the g.f. above, slightly rewritten. [Joerg Arndt, Mar 30 2014]

a(n) ~ 2^(n-1). - Vaclav Kotesovec, May 01 2014

a(n) = A098124(n)+A098125(n). - R. J. Mathar, Oct 01 2021

Extensions

More terms from Vladeta Jovovic, Sep 24 2004

Prepended a(0) = 1, Joerg Arndt, Mar 30 2014