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

1, 2, 6, 19, 51, 148, 395, 1095, 2945, 8020, 21597, 58518, 157746, 426250, 1149832, 3104236, 8375167, 22603530, 60988687, 164579663, 444082316, 1198312390, 3233419264, 8724918311, 23542640336, 63526028693, 171413973501, 462531951559, 1248062990751, 3367686427976

Offset: 1

Author

Olivier Gérard, Jul 30 2012

Keywords

• nonn
• nice

Comments

By "proper integer partition", one means that the case {n} is excluded for having only one part, equal to the number partitioned.

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]

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..2320 (first 70 terms from Vincenzo Librandi)

Crossrefs

Cf. A000041, A246828.

Programs

Maple

b:= proc(n, i) option remember; `if`(n<2, [1, n], `if`(i<1, 0,
      b(n, i-1)+(p-> p+[0, p[1]*a(i)])(b(n-i, min(n-i, i)))))
    end:
a:= n-> b(n, n-1)[2]:
seq(a(n), n=1..33);  # Alois P. Heinz, Dec 27 2023

Mathematica

Clear[a]; a[1] := 1; a[n_Integer] :=
a[n] = Plus @@ Map[Function[p, Plus @@ Map[a, p]], Drop[IntegerPartitions[n], 1]]; Table[ a[n], {n,1,30}]

Formula

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.

a(n) ~ exp(k) * a(n-1), k = 0.992632731... (conjecture). - Bill McEachen, Dec 26 2023