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A024382
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a(n) = n-th elementary symmetric function of the first n+1 positive integers congruent to 1 mod 4.
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3
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1, 6, 59, 812, 14389, 312114, 8011695, 237560280, 7990901865, 300659985630, 12511934225955, 570616907588100, 28301322505722525, 1516683700464669450, 87336792132539066775, 5378036128829898836400, 352652348707389385916625, 24533212082483855129037750
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OFFSET
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0,2
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COMMENTS
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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
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
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LINKS
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FORMULA
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a(n) = (8*n-2)*a(n-1) - (4*n-3)^2*a(n-2) for n>1. - Alois P. Heinz, Feb 25 2015
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EXAMPLE
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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
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MAPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(PARI) x = 'x + O('x^33); Vec(serlaplace((4-log(1-4*x))/(4*(1-4*x)^(5/4)))) \\ Gheorghe Coserea, Dec 24 2015
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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