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 A064986 Number of partitions of n into factorial parts (0! not allowed). 22
 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 (list; graph; refs; listen; history; text; internal format)
 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. 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 * A029019 A040039 A008667 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

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Last modified January 17 19:19 EST 2021. Contains 340247 sequences. (Running on oeis4.)