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%I #43 Dec 18 2022 07:08:10
%S 1,5,40,440,6160,104720,2094400,48171200,1252451200,36321084800,
%T 1162274713600,40679614976000,1545825369088000,63378840132608000,
%U 2788668965834752000,131067441394233344000,6553372069711667200000,347328719694718361600000,19450408302904228249600000
%N One half of triple factorial numbers.
%C Preface the series with a 1, then the next term = (1, 4, 7, 10, ...) dot (1, 1, 5, 40, ...). E.g., a(5) = 6160 = (1, 4, 7, 10, 13) dot (1, 1, 5, 40, 440) = (1 + 4 + 35 + 400 + 5720). - _Gary W. Adamson_, May 17 2010
%C In other words, a(n) = Sum_{i=0..n-1} b(i)*A016777(i) where b(0)=1 and b(n)=a(n). - _Michel Marcus_, Dec 18 2022
%H G. C. Greubel, <a href="/A034000/b034000.txt">Table of n, a(n) for n = 1..375</a>
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv:1403.5962 [math.CO], 2014.
%F a(n) = A007661(3n-1)/2 = A008544(n)/2.
%F 2*a(n+1) = (3*n+2)!!! = Product_{j=0..n} (3*j+2), n >= 0.
%F E.g.f.: (-1 + (1-3*x)^(-2/3))/2.
%F a(n) = (3*n-1)!/(2*3^(n-1)*(n-1)!*A007559(n)).
%F a(n) ~ 3/2*2^(1/2)*Pi^(1/2)*Gamma(2/3)^-1*n^(7/6)*3^n*e^-n*n^n*{1 + 23/36*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
%F a(n) = 3^n*(n+2/3)!/(2/3)!, with offset 0. - _Paul Barry_, Sep 04 2005
%F D-finite with recurrence a(n) + (1-3*n)*a(n-1) = 0. - _R. J. Mathar_, Dec 03 2012
%F Sum_{n>=1} 1/a(n) = 2*(e/3)^(1/3)*(Gamma(2/3) - Gamma(2/3, 1/3)). - _Amiram Eldar_, Dec 18 2022
%p A034000:=n->`if`(n=1, 1, (3*n-1)*A034000(n-1)); seq(A034000(n), n=1..20); # _G. C. Greubel_, Aug 15 2019
%t nxt[{n_,a_}]:={n+1,(3(n+1)-1)*a}; Transpose[NestList[nxt,{1,1},20]][[2]] (* _Harvey P. Dale_, Aug 22 2015 *)
%t Table[3^(n-1)*Pochhammer[5/3, n-1], {n,20}] (* _G. C. Greubel_, Aug 15 2019 *)
%o (PARI) m=20; v=concat([1], vector(m-1)); for(n=2, m, v[n]=(3*n-1)*v[n-1]); v \\ _G. C. Greubel_, Aug 15 2019
%o (Magma) [n le 1 select 1 else (3*n-1)*Self(n-1): n in [1..20]]; // _G. C. Greubel_, Aug 15 2019
%o (Sage)
%o def a(n):
%o if n==1: return 1
%o else: return (3*n-1)*a(n-1)
%o [a(n) for n in (1..20)] # _G. C. Greubel_, Aug 15 2019
%o (GAP) a:=[1];; for n in [2..20] do a[n]:=(3*n-1)*a[n-1]; od; a; # _G. C. Greubel_, Aug 15 2019
%Y Cf. A007559, A034001, A025748, A034724, A016777.
%K easy,nonn
%O 1,2
%A _Wolfdieter Lang_
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