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

0,3

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

Table of n, a(n) for n=0..43.

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

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

Sequence in context: A294102 A190969 A099451 * A133252 A299335 A247618

Adjacent sequences: A174791 A174792 A174793 * A174795 A174796 A174797

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