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A034687 Related to quintic factorial numbers A008548.

%I A034687 #29 Aug 17 2025 02:49:03
%S A034687 1,15,275,5500,115500,2502500,55412500,1246781250,28398906250,
%T A034687 653174843750,15141780468750,353308210937500,8289154179687500,
%U A034687 195387205664062500,4624163867382812500,109823891850341796875
%N A034687 Related to quintic factorial numbers A008548.
%C A034687 Convolution of A034688(n-1) with A025750(n), n >= 1.
%H A034687 Michael De Vlieger, <a href="/A034687/b034687.txt">Table of n, a(n) for n = 1..717</a> (first 500 terms from G. C. Greubel).
%H A034687 Wolfdieter Lang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of the Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
%H A034687 Elżbieta Liszewska and Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%F A034687 a(n) = 5^(n-1)*A008548(n)/n!, where A008548(n) = (5*n-4)(!^5) = Product_{j=1..n} (5*j-4).
%F A034687 G.f.: (-1 + (1-25*x)^(-1/5))/5.
%F A034687 E.g.f.: (1/5)*L_{-1/5}(25*x) - 1, where L_{k}(x) is the Laguerre polynomial. - _Stefano Spezia_, Aug 17 2019
%F A034687 a(n) ~ 5^(2*n-1) * n^(-4/5) / Gamma(1/5). - _Amiram Eldar_, Aug 17 2025
%p A034687 seq(5^(n-1)*(product(5*k+1, k = 0..n-1))/factorial(n), n = 1..20); # _G. C. Greubel_, Aug 17 2019
%t A034687 Table[5^(2*n-1)*Pochhammer[1/5, n]/n!, {n, 20}] (* _G. C. Greubel_, Aug 17 2019 *)
%o A034687 (PARI) vector(20, n, 5^(n-1)*prod(k=0, n-1, 5*k+1)/n!) \\ _G. C. Greubel_, Aug 17 2019
%o A034687 (Magma) [5^(n-1)*(&*[5*k+1: k in [0..n-1]])/Factorial(n): n in [1..20]]; // _G. C. Greubel_, Aug 17 2019
%o A034687 (Sage) [5^(n-1)*product(5*k+1 for k in (0..n-1))/factorial(n) for n in (1..20)] # _G. C. Greubel_, Aug 17 2019
%o A034687 (GAP) List([1..20], n-> 5^(n-1)*Product([0..n-1], k-> 5*k+1)/Factorial(n)); # _G. C. Greubel_, Aug 17 2019
%Y A034687 Cf. A008548, A025750, A034255, A034688, A175380.
%K A034687 easy,nonn
%O A034687 1,2
%A A034687 _Wolfdieter Lang_