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A320817 Number of partitions of n with exactly four sorts of part 1 which are introduced in ascending order.

%I A320817 #7 Dec 17 2020 12:12:15
%S A320817 1,10,66,361,1778,8207,36310,156095,657785,2733065,11241497,45900679,
%T A320817 186420826,754165809,3042167236,12245294090,49211278321,197535872510,
%U A320817 792216674789,3175088068035,12719020008668,50932090504830,203896407951944,816089798651203
%N A320817 Number of partitions of n with exactly four sorts of part 1 which are introduced in ascending order.
%H A320817 Alois P. Heinz, <a href="/A320817/b320817.txt">Table of n, a(n) for n = 4..1663</a>
%F A320817 a(n) = A320735(n) - A320734(n).
%p A320817 b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
%p A320817       Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
%p A320817     end:
%p A320817 a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(4):
%p A320817 seq(a(n), n=4..35);
%t A320817 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 A320817 a[n_] := With[{k = 4}, b[n, n, k] - b[n, n, k-1]];
%t A320817 a /@ Range[4, 35] (* _Jean-François Alcover_, Dec 17 2020, after _Alois P. Heinz_ *)
%Y A320817 Column k=4 of A292746.
%Y A320817 Cf. A320734, A320735.
%K A320817 nonn
%O A320817 4,2
%A A320817 _Alois P. Heinz_, Oct 21 2018