A174794 - OEIS

%I #14 Dec 29 2018 23:52:40

%S 0,1,5,17,45,97,181,305,477,705,997,1361,1805,2337,2965,3697,4541,

%T 5505,6597,7825,9197,10721,12405,14257,16285,18497,20901,23505,26317,

%U 29345,32597,36081,39805,43777,48005,52497,57261,62305,67637,73265,79197,85441,92005,98897

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

%C 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

%H Ronald Cools, <a href="http://nines.cs.kuleuven.be/ecf/">Encyclopaedia of Cubature Formulas</a>

%H Ronald Cools, <a href="https://doi.org/10.1016/S0377-0427(99)00229-0">Monomial cubature rules since "Stroud": a compilation - part 2</a>, Journal of Computational and Applied Mathematics - Numerical evaluation of integrals Vol. 112 (1999), 21-27.

%H Ronald Cools and Philip Rabinowitz, <a href="https://doi.org/10.1016/0377-0427(93)90027-9">Monomial cubature rules since "Stroud": a compilation</a>, Journal of Computational and Applied Mathematics Vol. 48 (1993), 309-326.

%H Paul Pichler, <a href="https://doi.org/10.1016/j.jedc.2010.09.009">Solving the multi-country Real Business Cycle model using a monomial rule Galerkin method</a>, Journal of Economic Dynamics and Control Vol. 35 (2011), 240-251.

%H Frank Stenger, <a href="http://www.jstor.org/stable/2004361">Tabulation of Certain Fully Symmetric Numerical Integration Formulas of Degree 3, 5, 7, 9, and 11</a>, Mathematics of Computation Vol. 25 (1971), 935.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F G.f.: x*(1 + x)*(1 + 3*x^2)/(1 - x)^4.

%F From _Franck Maminirina Ramaharo_, Dec 17 2018: (Start)

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 5.

%F a(n) = 8*binomial(n - 1, 3) + 8*binomial(n - 1, 2) + 4*binomial(n - 1, 1) + 1, n >= 1.

%F E.g.f.: (9 - (9 - 12*x - 4*x^3)*exp(x))/3. (End)

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

%o (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 */

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

%K nonn,easy

%O 0,3

%A _Roger L. Bagula_, Mar 29 2010

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