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Number of partitions of n having no parts with multiplicity 9.
8

%I #12 Apr 30 2018 12:30:45

%S 1,1,2,3,5,7,11,15,22,29,42,55,76,99,133,172,227,290,376,477,612,769,

%T 975,1217,1528,1895,2359,2907,3592,4400,5403,6584,8034,9742,11823,

%U 14272,17234,20713,24897,29803,35674,42542,50719,60272,71592,84794

%N Number of partitions of n having no parts with multiplicity 9.

%H Alois P. Heinz, <a href="/A184644/b184644.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A000041(n) - A183566(n).

%F a(n) = A183568(n,0) - A183568(n,9).

%F G.f.: Product_{j>0} (1-x^(9*j)+x^(10*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=9, [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 == 9, {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, A183566, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184643, A184645.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jan 18 2011