A064986 Number of partitions of n into factorial parts (0! not allowed).

1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 9, 9, 12, 12, 15, 15, 18, 18, 22, 22, 26, 26, 30, 30, 36, 36, 42, 42, 48, 48, 56, 56, 64, 64, 72, 72, 82, 82, 92, 92, 102, 102, 114, 114, 126, 126, 138, 138, 153, 153, 168, 168, 183, 183, 201, 201, 219, 219, 237, 237, 258, 258, 279, 279

Offset: 0

Author

Naohiro Nomoto, Oct 30 2001

Keywords

• easy
• nonn

Comments

a(2*n+1) = a(2*n) = A117930(n). [Reinhard Zumkeller, Dec 04 2011]

Examples

			a(3) = 2 because we can write 3 = 2!+1! = 1!+1!+1!.
a(10) = 9 because 10 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 = 1 + 1 + 1 + 1 + 1 + 1 + 2 + 2 = 1 + 1 + 1 + 1 + 2 + 2 + 2 = 1 + 1 + 2 + 2 + 2 + 2 = 2 + 2 + 2 + 2 + 2 = 1 + 1 + 1 + 1 + 6 = 1 + 1 + 2 + 6 = 2 + 2 + 6.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..250 from Reinhard Zumkeller)
• Youkow Homma, Jun Hwan Ryu and Benjamin Tong, Sequence non-squashing partitions, Slides from a talk, Jul 24 2014.
• Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A000142, A064985, A115944, A197182.

Bisection gives A090632.

Programs

Haskell

a064986 = p (tail a000142_list) where
   p _          0             = 1
   p fs'@(f:fs) m | m < f     = 0
                  | otherwise = p fs' (m - f) + p fs m
-- Reinhard Zumkeller, Dec 04 2011

Mathematica

b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i!>n, 0, b[n-i!, i]]];
c[n_] := Module[{i}, For[i = 1, i!<2n, i++]; b[2n, i]];
a[n_] := If[OddQ[n], c[(n-1)/2], c[n/2]];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 04 2020, after Alois P. Heinz in A117930 *)
Table[Length@IntegerPartitions[n, All, Factorial[Range[6]]], {n, 0, 63}] (* Robert Price, Jun 04 2020 *)

PARI

N=66; x='x+O('x^N); m=1; while(N>=m!, m++); Vec(1/prod(k=1, m-1, 1-x^k!)) \\ Seiichi Manyama, Oct 13 2019

PARI

N=66; x='x+O('x^N); m=1; while(N>=m!, m++); Vec(1+sum(i=1, m-1, x^i!/prod(j=1, i, 1-x^j!))) \\ Seiichi Manyama, Oct 13 2019

Formula

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

G.f.: 1 + Sum_{n>0} x^(n!) / Product_{k=1..n} (1 - x^(k!)). - Seiichi Manyama, Oct 12 2019

G.f.: 1 + x/(1-x) + x^2/((1-x)*(1-x^2)) + x^6/((1-x)*(1-x^2)*(1-x^6)) + ... . - Seiichi Manyama, Oct 12 2019

Extensions

More terms from Vladeta Jovovic and Don Reble, Nov 02 2001