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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.

%I A117930 #29 Nov 03 2023 16:05:08
%S A117930 1,2,3,5,7,9,12,15,18,22,26,30,36,42,48,56,64,72,82,92,102,114,126,
%T A117930 138,153,168,183,201,219,237,258,279,300,324,348,372,400,428,456,488,
%U A117930 520,552,588,624,660,700,740,780,825,870,915,965,1015,1065,1120,1175,1230
%N 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.
%C A117930 a(n) = A064986(2n) = A064986(2n+1). The first 48 terms of this sequence agree with those of A090632.
%C A117930 a(n) = A064986(2*n) = A064986(2*n+1). - _Reinhard Zumkeller_, Dec 04 2011
%H A117930 Alois P. Heinz, <a href="/A117930/b117930.txt">Table of n, a(n) for n = 0..1000</a> (first 250 terms from Reinhard Zumkeller)
%H A117930 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A117930 G.f.: 1/((1-x)*Product_{j>=2} (1 - x^(j!/2))).
%e A117930 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].
%p A117930 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);
%p A117930 # second Maple program
%p A117930 b:= proc(n, i) option remember;
%p A117930       `if`(n=0 or i=1, 1, b(n, i-1)+
%p A117930       `if`(i!>n, 0, b(n-i!, i)))
%p A117930     end:
%p A117930 a:= proc(n) local i;
%p A117930       for i while(i!<2*n) do od;
%p A117930       b(2*n, i)
%p A117930     end:
%p A117930 seq(a(n), n=0..100);  # _Alois P. Heinz_, Jun 13 2012
%t A117930 f[n_] := Length@ IntegerPartitions[2 n, All, {1, 2, 6, 24, 120}]; Array[f, 57, 0] (* _Robert G. Wilson v_, Oct 02 2014 *)
%t A117930 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-François Alcover_, Jun 29 2015, after _Alois P. Heinz_ *)
%o A117930 (Haskell)
%o A117930 a117930 n = p (tail a000142_list) $ 2*n where
%o A117930    p _          0             = 1
%o A117930    p ks'@(k:ks) m | m < k     = 0
%o A117930                   | otherwise = p ks' (m - k) + p ks m
%o A117930 -- _Reinhard Zumkeller_, Dec 04 2011
%Y A117930 Cf. A064986, A090632.
%K A117930 nonn
%O A117930 0,2
%A A117930 _Emeric Deutsch_, Apr 04 2006
%E A117930 An incorrect g.f. was deleted by _N. J. A. Sloane_, Sep 16 2009