A117930 Number of partitions of 2n into factorial parts (0! not allowed, i.e., only one kind of 1 can be a part). Also number of partitions of 2n+1 into factorial parts.

1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 36, 42, 48, 56, 64, 72, 82, 92, 102, 114, 126, 138, 153, 168, 183, 201, 219, 237, 258, 279, 300, 324, 348, 372, 400, 428, 456, 488, 520, 552, 588, 624, 660, 700, 740, 780, 825, 870, 915, 965, 1015, 1065, 1120, 1175, 1230

Offset

0,2

Comments

a(n) = A064986(2n) = A064986(2n+1). The first 48 terms of this sequence agree with those of A090632.

a(n) = A064986(2*n) = A064986(2*n+1). - Reinhard Zumkeller, Dec 04 2011

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 250 terms from Reinhard Zumkeller)

Index entries for sequences related to factorial numbers

Formula

G.f.: 1/((1-x)*Product_{j>=2} (1 - x^(j!/2))).

Example

a(3) = 5 because the partitions of 6 into factorials are [6], [2,2,2], [2,2,1,1], [2,1,1,1,1] and [1,1,1,1,1,1].

Maple

g:=1/(1-x)/product(1-x^(j!/2), j=2..7): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=0..65);
# second Maple program
b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1, b(n, i-1)+
      `if`(i!>n, 0, b(n-i!, i)))
    end:
a:= proc(n) local i;
      for i while(i!<2*n) do od;
      b(2*n, i)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 13 2012

Mathematica

f[n_] := Length@ IntegerPartitions[2 n, All, {1, 2, 6, 24, 120}]; Array[f, 57, 0] (* Robert G. Wilson v, Oct 02 2014 *)
b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i!>n, 0, b[n-i!, i] ] ]; a[n_] := Module[{i}, For[i=1, i!<2*n, i++]; b[2*n, i]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 29 2015, after Alois P. Heinz *)

Prog

(Haskell)
a117930 n = p (tail a000142_list) $ 2*n where
   p _          0             = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Dec 04 2011

Crossrefs

Cf. A064986, A090632.

Sequence in context: A001840 A022794 A025693 * A090632 A022786 A005704

Adjacent sequences: A117927 A117928 A117929 * A117931 A117932 A117933

Keyword

nonn

Author

Emeric Deutsch, Apr 04 2006

Extensions

An incorrect g.f. was deleted by N. J. A. Sloane, Sep 16 2009

Status

approved