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A258459 Number of partitions of n into parts of exactly 4 sorts which are introduced in ascending order.

%I A258459 #11 Oct 15 2018 10:31:07
%S A258459 1,11,77,438,2216,10422,46731,202814,860586,3593561,14834956,60735095,
%T A258459 247155292,1001318246,4043482110,16288762319,65500024027,263035832734,
%U A258459 1055252430510,4230340216034,16949359882259,67881449170593,271777855641517,1087867649157513
%N A258459 Number of partitions of n into parts of exactly 4 sorts which are introduced in ascending order.
%H A258459 Alois P. Heinz, <a href="/A258459/b258459.txt">Table of n, a(n) for n = 4..1000</a>
%F A258459 a(n) ~ c * 4^n, where c = 1/(24*Product_{n>=1} (1-1/4^n)) = 1/(24*QPochhammer[1/4, 1/4]) = 1/(24*A100221) = 0.060514735102066542326446... . - _Vaclav Kotesovec_, Jun 01 2015
%p A258459 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A258459       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p A258459     end:
%p A258459 T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
%p A258459 a:= n-> T(n,4):
%p A258459 seq(a(n), n=4..35);
%t A258459 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]];
%t A258459 T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i/(i!*(k - i)!), {i, 0, k}];
%t A258459 Table[T[n, 4], {n, 4, 35}] (* _Jean-François Alcover_, May 25 2018, translated from Maple *)
%Y A258459 Column k=4 of A256130.
%Y A258459 Cf. A320546.
%K A258459 nonn
%O A258459 4,2
%A A258459 _Alois P. Heinz_, May 30 2015