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A258462 Number of partitions of n into parts of exactly 7 sorts which are introduced in ascending order.

%I A258462 #12 Dec 07 2020 14:00:43
%S A258462 1,29,492,6401,70880,704676,6490951,56524414,471750267,3810085912,
%T A258462 29989229859,231255237311,1754111872429,13128442913712,97189645384884,
%U A258462 713050007285941,5192646586465458,37581376345088462,270593146237918806,1939929376872664097
%N A258462 Number of partitions of n into parts of exactly 7 sorts which are introduced in ascending order.
%H A258462 Alois P. Heinz, <a href="/A258462/b258462.txt">Table of n, a(n) for n = 7..1000</a>
%F A258462 a(n) ~ c * 7^n, where c = 1/(7!*Product_{n>=1} (1-1/7^n)) = 1/(7!*QPochhammer[1/7, 1/7]) = 0.0002371101666331046535758625585353... . - _Vaclav Kotesovec_, Jun 01 2015
%p A258462 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A258462       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p A258462     end:
%p A258462 T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
%p A258462 a:= n-> T(n,7):
%p A258462 seq(a(n), n=7..30);
%t A258462 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 A258462 T[n_, k_] := Sum[b[n, n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}];
%t A258462 Table[T[n, 7], {n, 7, 30}] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A258462 Column k=7 of A256130.
%Y A258462 Cf. A320549.
%K A258462 nonn
%O A258462 7,2
%A A258462 _Alois P. Heinz_, May 30 2015