A174794 a(0) = 0 and a(n) = (4*n^3 - 12*n^2 + 20*n - 9)/3 for n >= 1.

0, 1, 5, 17, 45, 97, 181, 305, 477, 705, 997, 1361, 1805, 2337, 2965, 3697, 4541, 5505, 6597, 7825, 9197, 10721, 12405, 14257, 16285, 18497, 20901, 23505, 26317, 29345, 32597, 36081, 39805, 43777, 48005, 52497, 57261, 62305, 67637, 73265, 79197, 85441, 92005, 98897

Offset: 0

Author

Roger L. Bagula, Mar 29 2010

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000292, A005843, A046092, A130809, A161680.

Programs

Mathematica

CoefficientList[Series[x*(1 + x)*(1 + 3*x^2)/(1 - x)^4, {x, 0, 50}], x]

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 */

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)

Extensions

Definition replaced by polynomial - The Assoc. Eds. of the OEIS, Aug 10 2010