A229863 - OEIS

%I #14 Oct 02 2013 10:07:44

%S 1,1,2,3,4,6,10,13,20,27,38,51,71,92,125,163,214,276,360,457,588,743,

%T 942,1182,1487,1848,2306,2852,3527,4335,5331,6511,7958,9675,11754,

%U 14223,17198,20710,24928,29901,35828,42808,51099,60823,72333,85811,101686,120244,142036,167430,197170,231761

%N The number of integer partitions P of n such that either the length k of P is not a part or P has at least one part equal to 1 (or both).

%F For n>=2; a(n) = A000041(n) - A008483(n-1)

%F G.f.: 1/E(x) - x*(1-x)*(1-x^2)/E(x) + x where E(x) = prod(k>=1, 1-x^k ).

%e a(10) = 38 because there are 42 unrestricted partitions of 10.  All of these except the following four are counted by this sequence: 8+2,5+3+2,4+3+3,4+2+2+2.

%t nn=51;a=Product[1/(1-x^k),{k,1,nn}];b= x(Product[1/(1-x^k),{k,3,nn}]-1);CoefficientList[Series[a-b,{x,0,nn}],x]

%o (PARI) x='x+O('x^66); Vec( 1/eta(x) - x*(1-x)*(1-x^2)/eta(x) + x )

%Y Cf. A229816.

%K nonn

%O 0,3

%A _Geoffrey Critzer_ and _Joerg Arndt_, Oct 01 2013