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A258463 Number of partitions of n into parts of exactly 8 sorts which are introduced in ascending order.

%I A258463 #12 Dec 07 2020 14:00:32
%S A258463 1,37,788,12705,172520,2084836,23169639,241881526,2406802476,
%T A258463 23064505721,214505275665,1947297442670,17332491414616,
%U A258463 151788374231505,1311496639250495,11205023121304298,94832831557086797,796244028801983324,6640545376656071546
%N A258463 Number of partitions of n into parts of exactly 8 sorts which are introduced in ascending order.
%H A258463 Alois P. Heinz, <a href="/A258463/b258463.txt">Table of n, a(n) for n = 8..1000</a>
%F A258463 a(n) ~ c * 8^n, where c = 1/(8!*Product_{n>=1} (1-1/8^n)) = 1/(8!*QPochhammer[1/8, 1/8]) = 0.0000288589880256565005640019500910465339603... . - _Vaclav Kotesovec_, Jun 01 2015
%p A258463 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A258463       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p A258463     end:
%p A258463 T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
%p A258463 a:= n-> T(n,8):
%p A258463 seq(a(n), n=8..30);
%t A258463 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 A258463 T[n_, k_] := Sum[b[n, n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}];
%t A258463 Table[T[n, 8], {n, 8, 30}] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A258463 Column k=8 of A256130.
%Y A258463 Cf. A320550.
%K A258463 nonn
%O A258463 8,2
%A A258463 _Alois P. Heinz_, May 30 2015