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

0,3

Comments

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

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

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

Example

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.

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 *)

Prog

(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
(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

Crossrefs

Cf. A000142, A064985, A115944, A197182.

Bisection gives A090632.

Sequence in context: A309685 A309683 A283529 * A349675 A029019 A040039

Adjacent sequences: A064983 A064984 A064985 * A064987 A064988 A064989

Keyword

easy,nonn

Author

Naohiro Nomoto, Oct 30 2001

Extensions

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

Status

approved