cp's OEIS Frontend

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

%I A064985 #32 Jun 04 2020 19:31:02
%S A064985 1,2,4,6,9,12,17,22,29,36,45,54,66,78,93,108,126,144,166,188,214,240,
%T A064985 270,300,336,372,414,456,504,552,608,664,728,792,864,936,1018,1100,
%U A064985 1192,1284,1386,1488,1602,1716,1842,1968,2106,2244,2397,2550,2718,2886
%N A064985 Number of partitions of n into factorial parts ( 0! allowed ).
%H A064985 Alois P. Heinz, <a href="/A064985/b064985.txt">Table of n, a(n) for n = 0..10000</a>
%F A064985 G.f.: 1/(Product_{i>=1} (1-x^(i!)))/(1-x).
%F A064985 G.f.: 1/(1-x) + Sum_{n>0} x^(n!) / Product_{k=0..n} (1 - x^(k!)). - _Seiichi Manyama_, Oct 12 2019
%e A064985 a(3) = 6 because we can write 3 = 2!+1! = 2!+0! = 1!+1!+1! = 0!+0!+0! = 1!+1!+0! = 1!+0!+0!.
%e A064985 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
%t A064985 nn=51;CoefficientList[Series[Product[1/(1-x^(i!)),{i,0,10}],{x,0,nn}],x] (* _Geoffrey Critzer_, Sep 29 2013 *)
%t A064985 Table[Length@IntegerPartitions[n, All, Factorial[Range[0, 6]]], {n, 0, 51}] (* _Robert Price_, Jun 04 2020 *)
%o A064985 (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
%o A064985 (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
%Y A064985 Cf. A064986.
%K A064985 easy,nonn
%O A064985 0,2
%A A064985 _Naohiro Nomoto_, Oct 30 2001
%E A064985 More terms from _Vladeta Jovovic_ and _Don Reble_, Nov 02 2001