A320817 - OEIS

%I #7 Dec 17 2020 12:12:15

%S 1,10,66,361,1778,8207,36310,156095,657785,2733065,11241497,45900679,

%T 186420826,754165809,3042167236,12245294090,49211278321,197535872510,

%U 792216674789,3175088068035,12719020008668,50932090504830,203896407951944,816089798651203

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

%H Alois P. Heinz, <a href="/A320817/b320817.txt">Table of n, a(n) for n = 4..1663</a>

%F a(n) = A320735(n) - A320734(n).

%p b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(

%p       Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))

%p     end:

%p a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(4):

%p seq(a(n), n=4..35);

%t 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 a[n_] := With[{k = 4}, b[n, n, k] - b[n, n, k-1]];

%t a /@ Range[4, 35] (* _Jean-François Alcover_, Dec 17 2020, after _Alois P. Heinz_ *)

%Y Column k=4 of A292746.

%Y Cf. A320734, A320735.

%K nonn

%O 4,2

%A _Alois P. Heinz_, Oct 21 2018