A077229 - OEIS

%I #20 Oct 01 2021 11:41:43

%S 1,1,1,3,5,11,23,48,98,204,421,863,1766,3606,7341,14913,30233,61175,

%T 123589,249344,502443,1011366,2033894,4086975,8206833,16469875,

%U 33035611,66234372,132745859,265961487,532717894,1066778687,2135822457,4275459730,8557335141,17125445575,34268965676,68568213419,137187103849,274458924246

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

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F G.f.: 1 + sum(k>=0, ((x^(k+1)-x)/(x-1))^k ). - _Vladeta Jovovic_, Sep 24 2004

%F 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]

%F a(n) ~ 2^(n-1). - _Vaclav Kotesovec_, May 01 2014

%F a(n) = A098124(n)+A098125(n). - _R. J. Mathar_, Oct 01 2021

%e 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).

%t 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 *)

%Y Row sums of A077227.

%Y Cf. A064174, A348125.

%K nonn

%O 0,4

%A _Henry Bottomley_, Oct 29 2002

%E More terms from _Vladeta Jovovic_, Sep 24 2004

%E Prepended a(0) = 1, _Joerg Arndt_, Mar 30 2014