login
Number of partitions of n into two sorts of parts having exactly 5 parts of the second sort.
2

%I #7 Dec 11 2020 05:59:27

%S 1,7,29,92,247,590,1291,2642,5114,9475,16882,29159,48914,80160,128405,

%T 201958,311878,474540,711222,1052949,1539214,2226774,3186866,4520230,

%U 6351925,8856519,12248350,16822982,22940863,31092121,41871405,56078299,74676693,98949447

%N Number of partitions of n into two sorts of parts having exactly 5 parts of the second sort.

%H Alois P. Heinz, <a href="/A258475/b258475.txt">Table of n, a(n) for n = 5..1000</a>

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

%p `if`(i<1, 0, add(b(n-i*j, i-1)*add(x^t*

%p binomial(j, t), t=0..min(5, j)), j=0..n/i))), x, 6)

%p end:

%p a:= n-> coeff(b(n$2), x, 5):

%p seq(a(n), n=5..40);

%t b[n_, i_] := b[n, i] = Series[If[n==0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]*Sum[ x^t*Binomial[j, t], {t, 0, Min[5, j]}], {j, 0, n/i}]]], {x, 0, 6}];

%t a[n_] := Coefficient[b[n, n], x, 5];

%t a /@ Range[5, 40] (* _Jean-François Alcover_, Dec 11 2020, after _Alois P. Heinz_ *)

%Y Column k=5 of A256193.

%K nonn

%O 5,2

%A _Alois P. Heinz_, May 31 2015