A064985 Number of partitions of n into factorial parts ( 0! allowed ).

1, 2, 4, 6, 9, 12, 17, 22, 29, 36, 45, 54, 66, 78, 93, 108, 126, 144, 166, 188, 214, 240, 270, 300, 336, 372, 414, 456, 504, 552, 608, 664, 728, 792, 864, 936, 1018, 1100, 1192, 1284, 1386, 1488, 1602, 1716, 1842, 1968, 2106, 2244, 2397, 2550, 2718, 2886

Offset: 0

Author

Naohiro Nomoto, Oct 30 2001

Keywords

• easy
• nonn

Examples

			a(3) = 6 because we can write 3 = 2!+1! = 2!+0! = 1!+1!+1! = 0!+0!+0! = 1!+1!+0! = 1!+0!+0!.
G.f.: 1/(1-x) + x/(1-x)^2 + x^2/((1-x)^2*(1-x^2)) + x^6/((1-x)^2*(1-x^2)*(1-x^6)) + ... . - _Seiichi Manyama_, Oct 12 2019
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A064986.

Programs

Mathematica

nn=51;CoefficientList[Series[Product[1/(1-x^(i!)),{i,0,10}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 29 2013 *)
Table[Length@IntegerPartitions[n, All, Factorial[Range[0, 6]]], {n, 0, 51}] (* Robert Price, Jun 04 2020 *)

PARI

N=66; x='x+O('x^N); m=1; while(N>=m!, m++); Vec(1/prod(k=0, 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/(1-x)+sum(i=1, m-1, x^i!/prod(j=0, i, 1-x^j!))) \\ Seiichi Manyama, Oct 13 2019

Formula

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

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

Extensions

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