OFFSET
0,3
COMMENTS
For n >= 1, a(n+1) = (4*n^3 + 8*n + 3)/3 is the number of evaluation points on the n-dimensional cube in Stenger's degree 7 cubature rule. - Franck Maminirina Ramaharo, Dec 18 2018
LINKS
Ronald Cools, Encyclopaedia of Cubature Formulas
Ronald Cools, Monomial cubature rules since "Stroud": a compilation - part 2, Journal of Computational and Applied Mathematics - Numerical evaluation of integrals Vol. 112 (1999), 21-27.
Ronald Cools and Philip Rabinowitz, Monomial cubature rules since "Stroud": a compilation, Journal of Computational and Applied Mathematics Vol. 48 (1993), 309-326.
Paul Pichler, Solving the multi-country Real Business Cycle model using a monomial rule Galerkin method, Journal of Economic Dynamics and Control Vol. 35 (2011), 240-251.
Frank Stenger, Tabulation of Certain Fully Symmetric Numerical Integration Formulas of Degree 3, 5, 7, 9, and 11, Mathematics of Computation Vol. 25 (1971), 935.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
G.f.: x*(1 + x)*(1 + 3*x^2)/(1 - x)^4.
From Franck Maminirina Ramaharo, Dec 17 2018: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 5.
a(n) = 8*binomial(n - 1, 3) + 8*binomial(n - 1, 2) + 4*binomial(n - 1, 1) + 1, n >= 1.
E.g.f.: (9 - (9 - 12*x - 4*x^3)*exp(x))/3. (End)
MATHEMATICA
CoefficientList[Series[x*(1 + x)*(1 + 3*x^2)/(1 - x)^4, {x, 0, 50}], x]
PROG
(Maxima) a[0] : 0$ a[n] := (4*n^3 - 12*n^2 + 20*n - 9)/3$ makelist(a[n], n, 0, 50); /* Martin Ettl, Oct 21 2012 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula, Mar 29 2010
EXTENSIONS
Definition replaced by polynomial - The Assoc. Eds. of the OEIS, Aug 10 2010
STATUS
approved