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A117222 Number of partitions of 3-smooth numbers into 3-smooth numbers.

%I A117222 #21 Oct 13 2021 19:28:32
%S A117222 1,2,3,5,10,18,23,53,128,194,586,947,2070,3667,16762,33036,93402,
%T A117222 200626,445869,1517300,3715025,14526494,39369076,111448616,541299314,
%U A117222 1713653236,5690596129,9832997667,35075665070,131672592907,973547058482,4162255238584,18810021557460
%N A117222 Number of partitions of 3-smooth numbers into 3-smooth numbers.
%H A117222 David A. Corneth, <a href="/A117222/b117222.txt">Table of n, a(n) for n = 1..200</a> (first 100 terms from Andrew Howroyd)
%F A117222 a(n) = A105420(A003586(n)).
%t A117222 is[n_] := n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3] == 1;
%t A117222 MkS[n_] := Module[{A = {}, i = 0}, While[Length[A] < n, i++; If[is[i], AppendTo[A, i]]]; A];
%t A117222 seq[n_] := Module[{A = MkS[n], p}, p = 1/Product[1 - x^A[[i]] + O[x]^(1 + A[[Length[A]]]), {i, Length[A]}] // Normal; Table[Coefficient[p, x, A[[i]]], {i, Length[A]}]];
%t A117222 seq[33] (* _Jean-François Alcover_, Oct 13 2021, after _Andrew Howroyd_ *)
%o A117222 (PARI) \\ here is(n) is test for A003586 inclusion.
%o A117222 is(n)={forprime(p=2, 3, n/=p^valuation(n,p)); n==1}
%o A117222 MkS(n)={my(A=List(), i=0); while(#A<n, i++; if(is(i), listput(A,i))); Vec(A)}
%o A117222 seq(n)={my(A=MkS(n), p=1/prod(i=1, #A, 1 - x^A[i] + O(x*x^A[#A]))); vector(#A, i, polcoef(p,A[i]))} \\ _Andrew Howroyd_, Jan 02 2020
%Y A117222 Cf. A003586, A090184, A105420, A117220, A117221.
%K A117222 nonn
%O A117222 1,2
%A A117222 _Reinhard Zumkeller_, Mar 04 2006
%E A117222 Terms a(23) and beyond from _Andrew Howroyd_, Jan 02 2020