cp's OEIS Frontend

A320816 Number of partitions of n with exactly three sorts of part 1 which are introduced in ascending order.

%I A320816 #11 Dec 16 2020 08:46:12
%S A320816 1,6,26,97,334,1095,3482,10855,33405,101925,309237,934691,2818110,
%T A320816 8482505,25504000,76625146,230101961,690759226,2073184749,6221368879,
%U A320816 18667736528,56010470158,168045932624,504166843427,1512558622966,4537792056226,13613608545770
%N A320816 Number of partitions of n with exactly three sorts of part 1 which are introduced in ascending order.
%H A320816 Alois P. Heinz, <a href="/A320816/b320816.txt">Table of n, a(n) for n = 3..2097</a>
%F A320816 a(n) = A320734(n) - A320733(n).
%p A320816 b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
%p A320816       Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
%p A320816     end:
%p A320816 a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(3):
%p A320816 seq(a(n), n=3..35);
%t A320816 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 A320816 a[n_] := With[{k = 3}, b[n, n, k] - b[n, n, k - 1]];
%t A320816 a /@ Range[3, 35] (* _Jean-François Alcover_, Dec 16 2020, after _Alois P. Heinz_ *)
%Y A320816 Column k=3 of A292746.
%Y A320816 Cf. A320733, A320734.
%K A320816 nonn
%O A320816 3,2
%A A320816 _Alois P. Heinz_, Oct 21 2018