%I #16 Dec 05 2020 05:51:45

%S 1,4,20,84,356,1444,5876,23604,94852,379908,1521492,6088148,24360548,

%T 97451492,389838708,1559394356,6237711300,24951007620,99804576340,

%U 399218968084,1596878076132,6387515000292,25550068873908,102200286367156,408801181153476

%N Number of partitions of n into 4 sorts of parts.

%H Alois P. Heinz, <a href="/A246936/b246936.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Product_{i>=1} 1/(1-4*x^i).

%F a(n) ~ c * 4^n, where c = Product_{k>=1} 1/(1-1/4^k) = A065446 * A132020 = 1.4523536424495970158347... . - _Vaclav Kotesovec_, Mar 19 2015

%F G.f.: Sum_{i>=0} 4^i*x^i/Product_{j=1..i} (1 - x^j). - _Ilya Gutkovskiy_, Apr 12 2018

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

%p b(n, i-1) +`if`(i>n, 0, 4*b(n-i, i))))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..25);

%t (O[x]^20 - 3/QPochhammer[4, x])[[3]] (* _Vladimir Reshetnikov_, Nov 20 2015 *)

%t b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1]+If[i>n, 0, 4 b[n-i, i]]]];

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

%t a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 05 2020, after _Alois P. Heinz_ *)

%Y Column k=4 of A246935.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Sep 08 2014