A133708 First differences of A047835.
0, 0, 1, 69, 1694, 22932, 208152, 1413720, 7697052, 35194302, 139687119, 493127635, 1577331756, 4637757488, 12679063488, 32529562560, 78917794128, 182184724908, 402332471541, 853769650041, 1747606106554, 3462012537060, 6656436729800, 12452933493000
Offset: 1
Keywords
Links
- T. D. Noe, Table of n, a(n) for n = 1..1003
Programs
-
Mathematica
Join[{0,0,1},Differences[Table[Product[Times@@((i+Range[4,7])/(i+Range[0,3])),{i,n}],{n,0,30}]]] (* Harvey P. Dale, Aug 08 2015 *)
Formula
In terms of Vandermonde determinants, a(n) = 1/864*sum {1 <= x_1,x_2,x_3 <= n} (x_1*x_2*x_3)^2*(det V(x_1,x_2,x_3))^2 = 1/864 *sum {1 <= i,j,k <= n} (i*j*k*(i-j)(i-k)(j-k))^2, where V(x_1,x_2,x_3) is the Vandermonde matrix of order 3. a(n) = (n-2)*(n-1)^2*n^3*(n+1)^3*(n+2)^2*(n+3)*(n^2+n+3)*(2n+1)/108864000.
Empirical G.f.: x^3*(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)/(x-1)^16. [Colin Barker, Jun 06 2012]