A184641 - OEIS

%I #11 Apr 30 2018 13:54:05

%S 1,1,2,3,5,7,10,15,21,29,40,54,72,96,127,166,216,279,358,457,580,735,

%T 924,1159,1446,1799,2228,2752,3388,4158,5087,6207,7551,9165,11093,

%U 13401,16144,19412,23286,27882,33310,39727,47289,56191,66647,78923,93299

%N Number of partitions of n having no parts with multiplicity 6.

%H Alois P. Heinz, <a href="/A184641/b184641.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A000041(n) - A183563(n).

%F a(n) = A183568(n,0) - A183568(n,6).

%F G.f.: Product_{j>0} (1-x^(6*j)+x^(7*j))/(1-x^j).

%e a(6) = 10, because 10 partitions of 6 have no parts with multiplicity 6: [1,1,1,1,2], [1,1,2,2], [2,2,2], [1,1,1,3], [1,2,3], [3,3], [1,1,4], [2,4], [1,5], [6].

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

%p       add((l->`if`(j=6, [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 == 6, {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, A183563, A183568, A007690, A116645, A118807, A184639, A184640, A184642, A184643, A184644, A184645.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jan 18 2011