%I #13 Sep 21 2024 13:41:40

%S 1,1,2,3,5,7,11,14,22,29,41,54,75,97,130,168,222,283,368,465,597,750,

%T 949,1183,1488,1841,2292,2822,3487,4267,5239,6376,7782,9429,11439,

%U 13798,16661,20007,24043,28763,34420,41021,48894,58066,68956,81627,96592

%N Number of partitions of n having no parts with multiplicity 7.

%H Alois P. Heinz, <a href="/A184642/b184642.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A000041(n) - A183564(n).

%F a(n) = A183568(n,0) - A183568(n,7).

%F G.f.: Product_{j>0} (1-x^(7*j)+x^(8*j))/(1-x^j).

%e a(7) = 14, because 14 partitions of 7 have no parts with multiplicity 7: [1,1,1,1,1,2], [1,1,1,2,2], [1,2,2,2], [1,1,1,1,3], [1,1,2,3], [2,2,3], [1,3,3], [1,1,1,4], [1,2,4], [3,4], [1,1,5], [2,5], [1,6], [7].

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

%p add((l->`if`(j=7, [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 == 7, {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_ *)

%t Table[Count[IntegerPartitions[n],_?(FreeQ[Length/@Split[#],7]&)],{n,0,50}] (* _Harvey P. Dale_, Sep 21 2024 *)

%Y Cf. A000041, A183564, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184643, A184644, A184645.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jan 18 2011