This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A021424 #35 May 07 2025 20:05:32
%S A021424 1,16,170,1520,12411,96096,719860,5278240,38153621,273134576,
%T A021424 1942326750,13748476560,97001079631,682818667456,4798793396840,
%U A021424 33686888924480,236284962774441,1656378634646736,11606570499786130
%N A021424 Expansion of 1/((1-x)(1-3x)(1-5x)(1-7x)).
%C A021424 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
%H A021424 Vincenzo Librandi, <a href="/A021424/b021424.txt">Table of n, a(n) for n = 0..200</a>
%H A021424 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (16,-86,176,-105).
%F A021424 a(n) = (7^n- 3*5^n+ 3^(n+1)-1)/48. - Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 21 2001
%F A021424 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
%F A021424 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
%F A021424 G.f.: 1/((1-x)*(1-3*x)*(1-5*x)*(1-7*x)). See the name.
%F A021424 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
%e A021424 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
%t A021424 Table[(7^n - 3*5^n + 3^(n + 1) - 1)/48, {n, 3, 60}]
%t A021424 CoefficientList[Series[1 / ((1 - x) (1 - 3 x) (1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 09 2013 *)
%t A021424 LinearRecurrence[{16,-86,176,-105},{1,16,170,1520},30] (* _Harvey P. Dale_, May 26 2014 *)
%o A021424 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1-3*x)*(1-5*x)*(1-7*x)))); // _Vincenzo Librandi_, Jul 09 2013
%o A021424 (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
%o A021424 (PARI) x='x+O('x^99); Vec(1/((1-x)*(1-3*x)*(1-5*x)*(1-7*x))) \\ _Altug Alkan_, Oct 11 2017
%Y A021424 Cf. A039755 (column k=3), A016209.
%K A021424 nonn,easy
%O A021424 0,2
%A A021424 _N. J. A. Sloane_