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

Author

Emeric Deutsch, Apr 04 2006

Keywords

• nonn

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

Examples

			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].
		

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

Crossrefs

Cf. A064986, A090632.

Programs

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

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 *)

Formula

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

Extensions

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