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A117930 Number of partitions of 2n into factorial parts (0! not allowed, i.e., only one kind of 1 can be a part). Also number of partitions of 2n+1 into factorial parts. 2
1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 36, 42, 48, 56, 64, 72, 82, 92, 102, 114, 126, 138, 153, 168, 183, 201, 219, 237, 258, 279, 300, 324, 348, 372, 400, 428, 456, 488, 520, 552, 588, 624, 660, 700, 740, 780, 825, 870, 915, 965, 1015, 1065, 1120, 1175, 1230 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = A064986(2n) = A064986(2n+1). The first 48 terms of this sequence agree with those of A090632.

a(n) = A064986(2*n) = A064986(2*n+1). - Reinhard Zumkeller, Dec 04 2011

LINKS

Reinhard Zumkeller and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 250 terms from Reinhard Zumkeller)

Index entries for sequences related to factorial numbers

FORMULA

G.f.: 1/((1-x)*Product_{j>=2} (1 - x^(j!/2))).

EXAMPLE

a(3) = 5 because the partitions of 6 into factorials are [6], [2,2,2], [2,2,1,1], [2,1,1,1,1] and [1,1,1,1,1,1].

MAPLE

g:=1/(1-x)/product(1-x^(j!/2), j=2..7): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=0..65);

# second Maple program

b:= proc(n, i) option remember;

      `if`(n=0 or i=1, 1, b(n, i-1)+

      `if`(i!>n, 0, b(n-i!, i)))

    end:

a:= proc(n) local i;

      for i while(i!<2*n) do od;

      b(2*n, i)

    end:

seq(a(n), n=0..100);  # Alois P. Heinz, Jun 13 2012

MATHEMATICA

f[n_] := Length@ IntegerPartitions[2 n, All, {1, 2, 6, 24, 120}]; Array[f, 57, 0] (* Robert G. Wilson v, Oct 02 2014 *)

b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i!>n, 0, b[n-i!, i] ] ]; a[n_] := Module[{i}, For[i=1, i!<2*n, i++]; b[2*n, i]]; Table[a[n], {n, 0, 100}] (* Jean-Fran├žois Alcover, Jun 29 2015, after Alois P. Heinz *)

PROG

(Haskell)

a117930 n = p (tail a000142_list) $ 2*n where

   p _          0             = 1

   p ks'@(k:ks) m | m < k     = 0

                  | otherwise = p ks' (m - k) + p ks m

-- Reinhard Zumkeller, Dec 04 2011

CROSSREFS

Cf. A064986, A090632.

Sequence in context: A001840 A022794 A025693 * A090632 A022786 A005704

Adjacent sequences:  A117927 A117928 A117929 * A117931 A117932 A117933

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Apr 04 2006

EXTENSIONS

An incorrect g.f. was deleted by N. J. A. Sloane, Sep 16 2009

STATUS

approved

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Last modified February 26 03:10 EST 2021. Contains 341619 sequences. (Running on oeis4.)