login
Number of partitions of n having no parts with multiplicity 8.
8

%I #11 Apr 30 2018 12:30:54

%S 1,1,2,3,5,7,11,15,21,30,41,55,75,99,131,172,223,288,372,474,603,764,

%T 962,1206,1509,1876,2326,2878,3543,4351,5330,6506,7921,9623,11655,

%U 14085,16987,20434,24529,29392,35138,41930,49947,59381,70474,83512,98779

%N Number of partitions of n having no parts with multiplicity 8.

%H Alois P. Heinz, <a href="/A184643/b184643.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A000041(n) - A183565(n).

%F a(n) = A183568(n,0) - A183568(n,8).

%F G.f.: Product_{j>0} (1-x^(8*j)+x^(9*j))/(1-x^j).

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],

%p add((l->`if`(j=8, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))

%p end:

%p a:= n-> (l-> l[1]-l[2])(b(n, n)):

%p seq(a(n), n=0..50);

%t b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 8, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];

%t a[n_] := b[n, n][[1]] - b[n, n][[2]];

%t Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)

%Y Cf. A000041, A183565, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184644, A184645.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jan 18 2011