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A214948 a(n) is the sum over all proper integer partitions of n of the previous terms.

%I A214948 #20 Dec 27 2023 11:37:50
%S A214948 1,2,6,19,51,148,395,1095,2945,8020,21597,58518,157746,426250,1149832,
%T A214948 3104236,8375167,22603530,60988687,164579663,444082316,1198312390,
%U A214948 3233419264,8724918311,23542640336,63526028693,171413973501,462531951559,1248062990751,3367686427976
%N A214948 a(n) is the sum over all proper integer partitions of n of the previous terms.
%C A214948 By "proper integer partition", one means that the case {n} is excluded for having only one part, equal to the number partitioned.
%C A214948 The growth of this function is exponential a(n) -> c * exp(n). [This is not correct, a(n) ~ c * d^n, where d = A246828 = 2.69832910647421123126399... and c = 0.39308289517441096263558422597609193642795355676880812197435683468376... - _Vaclav Kotesovec_, Dec 27 2023]
%H A214948 Alois P. Heinz, <a href="/A214948/b214948.txt">Table of n, a(n) for n = 1..2320</a> (first 70 terms from Vincenzo Librandi)
%F A214948 a(n) = sum( sum( a(i), i in p) , p in P*(n)) where P*(n) is the set of all integer partitions of n excluding {n}, p is a partition of P*(n), i is a part of p.
%F A214948 a(n) ~ exp(k) * a(n-1), k = 0.992632731... (conjecture). - _Bill McEachen_, Dec 26 2023
%e A214948 a(4) = (a(3)+a(1))+(a(2)+a(2))+(a(2)+a(1)+a(1))+(a(1)+a(1)+a(1)+a(1)) = (6 +  1) + (2 + 2) + (2 + 2*1) + (4*1) = 7 + 4 + 4 + 4 = 19.
%p A214948 b:= proc(n, i) option remember; `if`(n<2, [1, n], `if`(i<1, 0,
%p A214948       b(n, i-1)+(p-> p+[0, p[1]*a(i)])(b(n-i, min(n-i, i)))))
%p A214948     end:
%p A214948 a:= n-> b(n, n-1)[2]:
%p A214948 seq(a(n), n=1..33);  # _Alois P. Heinz_, Dec 27 2023
%t A214948 Clear[a]; a[1] := 1; a[n_Integer] :=
%t A214948 a[n] = Plus @@ Map[Function[p, Plus @@ Map[a, p]], Drop[IntegerPartitions[n], 1]]; Table[ a[n], {n,1,30}]
%Y A214948 Cf. A000041, A246828.
%K A214948 nonn,nice
%O A214948 1,2
%A A214948 _Olivier Gérard_, Jul 30 2012