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A047831 a(n) = Product_{i=1..n} ((i+5)*(i+6)*(i+7)*(i+8)*(i+9))/(i*(i+1)*(i+2)*(i+3)*(i+4)).

%I A047831 #30 Aug 31 2025 02:56:19
%S A047831 1,252,19404,731808,16818516,267227532,3184461423,30107635272,
%T A047831 235234907908,1566039386912,9095857138368,46960429261824,
%U A047831 218772384397632,931020034054176,3656383418054268,13365232267026024,45800747571406905,148055097314224100
%N A047831 a(n) = Product_{i=1..n} ((i+5)*(i+6)*(i+7)*(i+8)*(i+9))/(i*(i+1)*(i+2)*(i+3)*(i+4)).
%C A047831 Number of tilings of a <5,n,5> hexagon.
%D A047831 O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.
%H A047831 Seiichi Manyama, <a href="/A047831/b047831.txt">Table of n, a(n) for n = 0..10000</a>
%H A047831 O. D. Anderson, <a href="/A002415/a002415.pdf">Find the next sequence</a>, J. Rec. Math., 8 (No. 4, 1975-1976), 241. [Annotated scanned copy]
%H A047831 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 25.
%F A047831 a(n) = C(n+5,n+4)*C(n+6,n+3)*C(n+7,n+2)*C(n+8,n+1)*C(n+9,n)/(140*5!). - _Zerinvary Lajos_, May 29 2007
%F A047831 Sum_{n>=0} 1/a(n) = 876090600000*zeta(3) + 266716800000*zeta(5) - 5318707224821/4. - _Amiram Eldar_, Aug 31 2025
%p A047831 seq(binomial(n+5,n+4)*binomial(n+6,n+3)*binomial(n+7,n+2)*binomial(n+8,n+1)*binomial(n+9,n)/(140*5!), n=0..17); # _Zerinvary Lajos_, May 29 2007
%t A047831 Table[Product[Times@@(i+Range[5,9])/Times@@(i+Range[0,4]),{i,n}],{n,0,20}] (* _Harvey P. Dale_, Jan 30 2015 *)
%o A047831 (PARI) a(n) = prod(k=1, 5, binomial(n+k+4, n-k+5))/(140*5!); \\ _Seiichi Manyama_, Apr 02 2021
%Y A047831 Fifth row of array A103905.
%Y A047831 Cf. A002117, A013663.
%K A047831 nonn,changed
%O A047831 0,2
%A A047831 _N. J. A. Sloane_
%E A047831 Definition corrected by Daniel Soll (soll(AT)mathematik.uni-marburg.de), Aug 31 2004