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A320818 Number of partitions of n with exactly five sorts of part 1 which are introduced in ascending order.

%I A320818 #8 Dec 17 2020 14:57:59
%S A320818 1,15,141,1066,7108,43747,255045,1431320,7814385,41804990,220266447,
%T A320818 1147232914,5922585396,30367092789,154877631181,786633449995,
%U A320818 3982378528296,20109428513990,101339359244739,509871884291730,2562078441467318,12861324297841420
%N A320818 Number of partitions of n with exactly five sorts of part 1 which are introduced in ascending order.
%H A320818 Alois P. Heinz, <a href="/A320818/b320818.txt">Table of n, a(n) for n = 5..1433</a>
%F A320818 a(n) = A320736(n) - A320735(n).
%p A320818 b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
%p A320818       Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
%p A320818     end:
%p A320818 a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(5):
%p A320818 seq(a(n), n=5..35);
%t A320818 b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, k}], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
%t A320818 a[n_] := With[{k = 5}, b[n, n, k] - b[n, n, k - 1]];
%t A320818 a /@ Range[5, 35] (* _Jean-François Alcover_, Dec 17 2020, after _Alois P. Heinz_ *)
%Y A320818 Column k=5 of A292746.
%Y A320818 Cf. A320735, A320736.
%K A320818 nonn
%O A320818 5,2
%A A320818 _Alois P. Heinz_, Oct 21 2018