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%I #50 Nov 01 2020 06:07:58
%S 1,1,3,12,72,480,3780,35280,372960,4263840,54432000,758419200,
%T 11436163200,185253868800,3214699488000,59172265152000,
%U 1163830187520000,24097823253504000,525794940582912000,12073276215576576000,290883846352619520000,7318777466097377280000
%N E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!).
%H Seiichi Manyama, <a href="/A293138/b293138.txt">Table of n, a(n) for n = 0..444</a>
%F a(n) ~ c^(1/4) * exp(2*sqrt(c*n) - n) * n^(n+1/2) / (sqrt(5) * n^(3/4)), where c = -polylog(2, -1/2 - I/2) - polylog(2, -1/2 + I/2) = 0.9669456127221570300837545... Equivalently, c = -Sum_{k>=1} (-1)^k * cos(Pi*k/4) / (k^2 * 2^(k/2-1)). - _Vaclav Kotesovec_, Oct 01 2017
%e Let's consider the partitions of n where no positive integer appears more than twice. (See A000726)
%e For n = 5,
%e partition | |
%e --------------------------------------------------------------------
%e 5 -> one 5 -> 1/(1!) (= 1 )
%e = 4 + 1 -> one 4 and one 1 -> 1/(1!*1!) (= 1 )
%e = 3 + 2 -> one 3 and one 2 -> 1/(1!*1!) (= 1 )
%e = 3 + 1 + 1 -> one 3 and two 1 -> 1/(1!*2!) (= 1/2)
%e = 2 + 2 + 1 -> two 2 and one 1 -> 1/(2!*1!) (= 1/2)
%e --------------------------------------------------------------------
%e sum 4
%e So a(5) = 5! * 4 = 480.
%e For n = 6,
%e partition | |
%e --------------------------------------------------------------------
%e 6 -> one 6 -> 1/(1!) (= 1 )
%e = 5 + 1 -> one 5 and one 1 -> 1/(1!*1!) (= 1 )
%e = 4 + 2 -> one 4 and one 2 -> 1/(1!*1!) (= 1 )
%e = 4 + 1 + 1 -> one 4 and two 1 -> 1/(1!*2!) (= 1/2)
%e = 3 + 3 -> two 3 -> 1/(2!) (= 1/2)
%e = 3 + 2 + 1 -> one 3, one 2 and one 1 -> 1/(1!*1!*1!) (= 1 )
%e = 2 + 2 + 1 + 1 -> two 2 and two 1 -> 1/(2!*2!) (= 1/4)
%e --------------------------------------------------------------------
%e sum 21/4
%e So a(6) = 6! * 21/4 = 3780.
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(b(n-i*j, i-1)/j!, j=0..min(2, n/i))))
%p end:
%p a:= n-> n!*b(n$2):
%p seq(a(n), n=0..23); # _Alois P. Heinz_, Oct 02 2017
%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1]/j!, {j, 0, Min[2, n/i]}]]];
%t a[n_] := n! b[n, n];
%t a /@ Range[0, 23] (* _Jean-François Alcover_, Nov 01 2020, after _Alois P. Heinz_ *)
%Y Column k=2 of A293135.
%Y Cf. A000726, A162891, A263401, A293141.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Oct 01 2017
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