A047835 a(n) = Product_{i=1..n} ((i+4)*(i+5)*(i+6)*(i+7))/(i*(i+1)*(i+2)*(i+3)).

1, 70, 1764, 24696, 232848, 1646568, 9343620, 44537922, 184225041, 677352676, 2254684432, 6892441920, 19571505408, 52101067968, 131018862096, 313203587004, 715536058545, 1569305708586, 3316911815140, 6778924352200, 13435361082000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of tilings of a <4,n,4> hexagon.

Partial sums of A133708. - Peter Bala, Sep 21 2007

References

• O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241. [Annotated scanned copy]
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 25.

Crossrefs

Fourth row of array A103905.

Cf. A002415, A047819, A133708.

Programs

Maple

seq(binomial(n,n-1)*binomial(n+1,n-2)*binomial(n+2,n-3)*binomial(n+3,n-4)/(10*4!), n=4..24); # Zerinvary Lajos, May 29 2007

Mathematica

Table[Product[Times@@((i+Range[4,7])/(i+Range[0,3])),{i,n}],{n,0,20}] (* Harvey P. Dale, Nov 03 2011 *)

Formula

a(n) = C(n,n-1)*C(n+1,n-2)*C(n+2,n-3)*C(n+3,n-4)/(10*4!), n >= 4 . - Zerinvary Lajos, May 29 2007

a(n-4) = (1/3456)*Sum_{1 <= x_1, x_2, x_3, x_4 <= n} (det V(x_1,x_2,x_3,x_4))^2 = (1/3456)*Sum_{1 <= i,j,k,l <= n} ((i-j)(i-k)(i-l)(j-k)(j-l)(k-l))^2, where V(x_1,x_2,x_3,x_4) is the Vandermonde matrix of order 4. - Peter Bala, Sep 21 2007

Empirical g.f.: (x+1)*(x^8 + 52*x^7 + 658*x^6 + 2890*x^5 + 4810*x^4 + 2890*x^3 + 658*x^2 + 52*x + 1)/(1-x)^17. - Colin Barker, Jun 06 2012

Sum_{n>=0} 1/a(n) = 67200*Pi^4 + 5605600*Pi^2 - 185612833/3. - Amiram Eldar, May 29 2022