A006332 From the enumeration of corners. (Formerly M2148)

0, 2, 28, 168, 660, 2002, 5096, 11424, 23256, 43890, 77924, 131560, 212940, 332514, 503440, 742016, 1068144, 1505826, 2083692, 2835560, 3801028, 5026098, 6563832, 8475040, 10829000, 13704210, 17189172, 21383208, 26397308, 32355010, 39393312, 47663616, 57332704

Offset

0,2

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Germain Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle}, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = (n*(1 + n)^2*(2 + n)*(1 + 2*n)*(3 + 2*n))/90.

a(n) = 2*A006858(n).

a(n) = (-1)^(n+1)*A132339(3, n).

G.f.: 2*(1+x)*(1 + 6*x + x^2)/(1-x)^7.

From G. C. Greubel, Dec 14 2021: (Start)

E.g.f.: (1/90)*x*(180 + 1080*x + 1350*x^2 + 555*x^3 + 84*x^4 + 4*x^5)*exp(x).

a(n) = binomial(n+2, 3)*binomial(2*n+3, 3)/5. (End)

From Amiram Eldar, Jul 10 2023: (Start)

Sum_{n>=1} 1/a(n) = 15*Pi^2 - 295/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = -15*Pi^2/2 + 120*Pi - 605/2. (End)

Maple

A006332:=-2*(1+z)*(z**2+6*z+1)/(z-1)**7; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Table[(n(1+n)^2(2+n)(1+2n)(3+2n))/90, {n, 0, 30}] (* or *)
{0}~Join~CoefficientList[Series[2(x+1)(x^2 +6x +1)/(1-x)^7, {x, 0, 29}], x] (* Michael De Vlieger, Mar 26 2016 *)

Prog

(PARI) my(x='x+O('x^99)); concat(0, Vec(2*(x+1)*(x^2+6*x+1)/(1-x)^7)) \\ Altug Alkan, Mar 26 2016
(Magma) [Binomial(n+2, 3)*Binomial(2*n+3, 3)/5: n in [0..30]]; // G. C. Greubel, Dec 14 2021
(Sage) [binomial(n+2, 3)*binomial(2*n+3, 3)/5 for n in (0..30)] # G. C. Greubel, Dec 14 2021

Crossrefs

Cf. A006858, A132339.

Sequence in context: A110241 A338426 A347602 * A280120 A065340 A001798

Adjacent sequences: A006329 A006330 A006331 * A006333 A006334 A006335

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved