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A064986
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Number of partitions of n into factorial parts (0! not allowed).
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22
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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
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OFFSET
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0,3
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COMMENTS
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a(2*n+1) = a(2*n) = A117930(n). [Reinhard Zumkeller, Dec 04 2011]
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000142, A064985, A115944, A197182.
Bisection gives A090632.
Sequence in context: A309685 A309683 A283529 * A029019 A040039 A008667
Adjacent sequences: A064983 A064984 A064985 * A064987 A064988 A064989
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KEYWORD
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easy,nonn
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AUTHOR
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Naohiro Nomoto, Oct 30 2001
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EXTENSIONS
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More terms from Vladeta Jovovic and Don Reble, Nov 02 2001
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STATUS
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approved
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