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%I #38 Sep 08 2022 08:44:48
%S 1,6,59,812,14389,312114,8011695,237560280,7990901865,300659985630,
%T 12511934225955,570616907588100,28301322505722525,1516683700464669450,
%U 87336792132539066775,5378036128829898836400,352652348707389385916625,24533212082483855129037750
%N a(n) = n-th elementary symmetric function of the first n+1 positive integers congruent to 1 mod 4.
%C 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
%C From _R. J. Mathar_, Oct 01 2016: (Start)
%C 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:
%C 1
%C 1 1
%C 1 6 5
%C 1 15 59 45
%C 1 28 254 812 585
%C 1 45 730 5130 14389 9945
%C 1 66 1675 20460 122119 312114 208845
%C 1 91 3325 62335 633619 3365089 8011695 5221125
%C 1 120 5964 158760 2441334 21740040 105599276 237560280 151412625
%C This here is the first subdiagonal. The diagonal seems to be A007696. The 2nd column is A000384, the 3rd A024378, the 4th A024379. (End)
%H Alois P. Heinz, <a href="/A024382/b024382.txt">Table of n, a(n) for n = 0..350</a>
%F a(n) = (8*n-2)*a(n-1) - (4*n-3)^2*a(n-2) for n>1. - _Alois P. Heinz_, Feb 25 2015
%F E.g.f.: (4-log(1-4*x))/(4*(1-4*x)^(5/4)). - _Gheorghe Coserea_, Dec 24 2015
%e For n = 1 we have a(1) = 1*5*(1/1 + 1/5) = 6.
%e For n = 2 we have a(2) = 1*5*9*(1/1 + 1/5 + 1/9) = 59.
%e 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
%p a:= proc(n) option remember; `if`(n<3, [1, 6, 59][n+1],
%p (8*n-2)*a(n-1) -(4*n-3)^2*a(n-2))
%p end;
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Feb 25 2015
%t Table[Det[Array[KroneckerDelta[#1,#2]((4*#1+2)-1)+1&,{k, k}]],{k,1,10}] (* _John M. Campbell_, May 23 2011 *)
%t 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 *)
%o (PARI) x = 'x + O('x^33); Vec(serlaplace((4-log(1-4*x))/(4*(1-4*x)^(5/4)))) \\ _Gheorghe Coserea_, Dec 24 2015
%o (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
%Y Cf. A024216.
%K nonn
%O 0,2
%A _Clark Kimberling_
%E More terms from _Alois P. Heinz_, Feb 25 2015
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