A212549 - OEIS

%I #13 Jun 08 2022 18:53:31

%S 0,0,1,1,3,4,8,11,19,26,41,56,83,111,158,209,287,375,503,648,852,1086,

%T 1403,1770,2255,2817,3546,4393,5469,6723,8294,10120,12382,15011,18228,

%U 21965,26497,31749,38069,45383,54114,64204,76176,89975,106259,124998,146987

%N Number of partitions of n containing at least one part m-9 if m is the largest part.

%H Alois P. Heinz, <a href="/A212549/b212549.txt">Table of n, a(n) for n = 9..1000</a>

%F G.f.: Sum_{i>0} x^(2*i+9) / Product_{j=1..9+i} (1-x^j).

%e a(11) = 1: [10,1].

%e a(12) = 1: [10,1,1].

%e a(13) = 3: [10,1,1,1], [10,2,1], [11,2].

%e a(14) = 4: [10,1,1,1,1], [10,2,1,1], [10,3,1], [11,2,1].

%e a(15) = 8: [10,1,1,1,1,1], [10,2,1,1,1], [10,2,2,1], [10,3,1,1], [10,4,1], [11,2,1,1], [11,2,2], [12,3].

%p b:= proc(n, i) option remember;

%p       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))

%p     end:

%p a:= n-> add(b(n-2*m-9, min(n-2*m-9, m+9)), m=1..(n-9)/2):

%p seq(a(n), n=9..60);

%t b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];

%t a[n_] := Sum[b[n - 2 m - 9, Min[n - 2 m - 9, m + 9]], {m, 1, (n - 9)/2}];

%t a /@ Range[9, 60] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)

%t Table[Count[IntegerPartitions[n],_?(MemberQ[#,#[[1]]-9]&)],{n,9,60}] (* _Harvey P. Dale_, Jun 08 2022 *)

%Y Column k=9 of A212551.

%K nonn

%O 9,5

%A _Alois P. Heinz_, May 20 2012