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

1,2

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]

Links

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

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

Example

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.

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}]

Crossrefs

Cf. A000041, A246828.

Sequence in context: A034533 A192711 A192712 * A192713 A014559 A192714

Adjacent sequences: A214945 A214946 A214947 * A214949 A214950 A214951

Keyword

nonn,nice

Author

Olivier Gérard, Jul 30 2012

Status

approved