A024382 a(n) = n-th elementary symmetric function of the first n+1 positive integers congruent to 1 mod 4.

1, 6, 59, 812, 14389, 312114, 8011695, 237560280, 7990901865, 300659985630, 12511934225955, 570616907588100, 28301322505722525, 1516683700464669450, 87336792132539066775, 5378036128829898836400, 352652348707389385916625, 24533212082483855129037750

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

a(n) is equal to the determinant of the n X n matrix whose (i,j)-entry is KroneckerDelta[i,j]((4*i+2)-1)+1. - John M. Campbell, May 23 2011

From R. J. Mathar, Oct 01 2016: (Start)

The k-th elementary symmetric functions of the integers 1+4*j, j=1..n, form a triangle T(n,k), 0<=k<=n, n>=0:

1

1 1

1 6 5

1 15 59 45

1 28 254 812 585

1 45 730 5130 14389 9945

1 66 1675 20460 122119 312114 208845

1 91 3325 62335 633619 3365089 8011695 5221125

1 120 5964 158760 2441334 21740040 105599276 237560280 151412625

This here is the first subdiagonal. The diagonal seems to be A007696. The 2nd column is A000384, the 3rd A024378, the 4th A024379. (End)

Examples

			For n = 1 we have a(1) = 1*5*(1/1 + 1/5) = 6.
For n = 2 we have a(2) = 1*5*9*(1/1 + 1/5 + 1/9) = 59.
For n = 3 we have a(3) = 1*5*9*13*(1/1 + 1/5 + 1/9 + 1/13) = 812. - _Gheorghe Coserea_, Dec 24 2015
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..350

Crossrefs

Cf. A024216.

Programs

Magma

I:=[1,6]; [n le 2 select I[n] else (8*n-10)*Self(n-1)-(4*n-7)^2*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 26 2015

Maple

a:= proc(n) option remember; `if`(n<3, [1, 6, 59][n+1],
      (8*n-2)*a(n-1) -(4*n-3)^2*a(n-2))
    end;
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 25 2015

Mathematica

Table[Det[Array[KroneckerDelta[#1,#2]((4*#1+2)-1)+1&,{k, k}]],{k,1,10}] (* John M. Campbell, May 23 2011 *)
RecurrenceTable[{a[0] == 1, a[1] == 6, a[n] == (8 n - 2) a[n - 1] - (4 n - 3)^2 a[n - 2]}, a, {n, 0, 20}] (* Vincenzo Librandi, Dec 26 2015 *)

PARI

x = 'x + O('x^33); Vec(serlaplace((4-log(1-4*x))/(4*(1-4*x)^(5/4)))) \\ Gheorghe Coserea, Dec 24 2015

Formula

a(n) = (8*n-2)*a(n-1) - (4*n-3)^2*a(n-2) for n>1. - Alois P. Heinz, Feb 25 2015

E.g.f.: (4-log(1-4*x))/(4*(1-4*x)^(5/4)). - Gheorghe Coserea, Dec 24 2015

Extensions

More terms from Alois P. Heinz, Feb 25 2015