A212546 - OEIS

%I #11 Dec 07 2020 09:36:44

%S 0,0,1,1,3,4,8,11,19,26,40,54,79,104,146,190,257,330,436,552,715,896,

%T 1140,1415,1777,2184,2711,3308,4063,4922,5995,7214,8720,10435,12525,

%U 14910,17793,21076,25016,29507,34850,40941,48148,56351,66007,76995,89855,104484

%N Number of partitions of n containing at least one part m-6 if m is the largest part.

%H Alois P. Heinz, <a href="/A212546/b212546.txt">Table of n, a(n) for n = 6..1000</a>

%F G.f.: Sum_{i>0} x^(2*i+6) / Product_{j=1..6+i} (1-x^j).

%e a(8) = 1: [7,1].

%e a(9) = 1: [7,1,1].

%e a(10) = 3: [7,1,1,1], [7,2,1], [8,2].

%e a(11) = 4: [7,1,1,1,1], [7,2,1,1], [7,3,1], [8,2,1].

%e a(12) = 8: [7,1,1,1,1,1], [7,2,1,1,1], [7,2,2,1], [7,3,1,1], [7,4,1], [8,2,1,1], [8,2,2], [9,3].

%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-6, min(n-2*m-6, m+6)), m=1..(n-6)/2):

%p seq(a(n), n=6..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 - 6, Min[n - 2 m - 6, m + 6]], {m, 1, (n - 6)/2}];

%t a /@ Range[6, 60] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)

%Y Column k=6 of A212551.

%K nonn

%O 6,5

%A _Alois P. Heinz_, May 20 2012