A021424 Expansion of 1/((1-x)(1-3x)(1-5x)(1-7x)).

1, 16, 170, 1520, 12411, 96096, 719860, 5278240, 38153621, 273134576, 1942326750, 13748476560, 97001079631, 682818667456, 4798793396840, 33686888924480, 236284962774441, 1656378634646736, 11606570499786130

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is h^{(4)}n, the complete homogeneous symmetric function of the four symbols s_j = 1 + 2*j, j = 0..3, of degree n >= 1, with h^{(4)}_0 = 1. See an example below. Thus it is the (dimensionless) volume of all multichoose(4, n) = binomial(n+3, 3) polytopes of dimension n with side lengths from the set {1, 3, 5, 7}. - _Wolfdieter Lang, May 26 2017

Examples

			a(2) = h^{(4)}_2 = (1^2 + 3^2 + 5^2 + 7^2) +  (1^1*(3^1 + 5^1 + 7^1) + 3^1*(5^1 + 7^1) + 5^1*7^1)  = 84 + 86  = 120. - _Wolfdieter Lang_, May 26 2017
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (16,-86,176,-105).

Crossrefs

Cf. A039755 (column k=3), A016209.

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1-3*x)*(1-5*x)*(1-7*x)))); // Vincenzo Librandi, Jul 09 2013

Magma

I:=[1, 16, 170, 1520]; [n le 4 select I[n] else 16*Self(n-1)-86*Self(n-2)+176*Self(n-3)-105*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Jul 09 2013

Mathematica

Table[(7^n - 3*5^n + 3^(n + 1) - 1)/48, {n, 3, 60}]
CoefficientList[Series[1 / ((1 - x) (1 - 3 x) (1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 09 2013 *)
LinearRecurrence[{16,-86,176,-105},{1,16,170,1520},30] (* Harvey P. Dale, May 26 2014 *)

PARI

x='x+O('x^99); Vec(1/((1-x)*(1-3*x)*(1-5*x)*(1-7*x))) \\ Altug Alkan, Oct 11 2017

Formula

a(n) = (7^n- 3*5^n+ 3^(n+1)-1)/48. - Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 21 2001

a(n) = 12*a(n-1) - 35*a(n-2) + (3^n-1)/2 with a(0)=1, a(1)=16. - Vincenzo Librandi, Jul 09 2013

a(n) = 16*a(n-1) - 86*a(n-2) + 176*a(n-3) - 105*a(n-4), with a(0)=1, a(1)=16, a(2)=170, a(3)=1520. - Vincenzo Librandi, Jul 09 2013

G.f.: 1/((1-x)*(1-3*x)*(1-5*x)*(1-7*x)). See the name.

E.g.f.: (343*exp(7*x) - 375*exp(5*x) + 81*exp(3*x) - exp(x))/48, from the e.g.f. of the fourth column (k=3) of A039755. - Wolfdieter Lang, May 26 2017