A203245 -id:A203245 - cp's OEIS frontend

A203007 (n-1)-st elementary symmetric function of Fibonacci numbers F(2) to F(n).

1, 3, 11, 61, 518, 6974, 149574, 5151036, 285534660, 25535107140, 3687959921760, 860864908848480, 324911938205144160, 198334214378751672000, 195840008156732278248000, 312839537789862069432264000

Offset: 1

Clark Kimberling, Dec 29 2011

nonn

From R. J. Mathar, Oct 01 2016 (Start):
The k-th elementary symmetric functions of F(j), j=2..n+1, form a triangle T(n,k), 0<=k<=n, n>=0:
1
1 1
1 3 2
1 6 11 6
1 11 41 61 30
1 19 129 389 518 240
1 32 376 2066 5575 6974 3120
1 53 1048 9962 48961 124049 149574 65520
1 87 2850 45594 387669 1788723 4367240 5151036 2227680
This here is the first subdiagonal. The diagonal is A003266. The 2nd column is A001911, the 3rd A203245. (End)

Cf. A000045, A203006.

f[k_] := Fibonacci[k + 1]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A203007 *)

A379104 a(n) = third elementary symmetric function of the first n distinct Fibonacci numbers.

Original entry on oeis.org

6, 61, 389, 2066, 9962, 45594, 202344, 881859, 3801171, 16275292, 69399116, 295177196, 1253532482, 5318285553, 22550198601, 95580699774, 405034367814, 1716140731030, 7270703692340, 30801852323495, 130485697292231, 552764498063256, 2341595675572344

Offset: 3

Clark Kimberling, Dec 16 2024

nonn

Conjecture: for k >=1, the k-th elementary symmetric function (esf) of first n distinct Fibonacci numbers (fndFn) is a homogeneous linear recurrence sequence of order (n+2)(n+1)/2.
2nd esf of fndFn is given by A203245, with signature (4,-2,-6,4,2,-1).
3rd esf of fndFn is this sequence, with signature (7, -8, -27, 45, 24, -51, -3, 16, -1, -1).
4th esf of fndFn has signature (12,-28,-107,400,120,-1298,440,1408,-726,-560,296,83,-36,-4,1).
5th esf of fndFn has signature (20,-84,-423,3056,8,-25893,25956,73892,-105763,-77952,146160,30653,-86884,-3276,23499,-496,-2680,105,108,-4,-1).
The 6th esf of fndFn has signature (33,-240,-1671,22231,-12264,-477708,1054788,3271080,-10808292,-6412404,40815192,-4411686,-71500002,25737096,64629222,-28878366,-31047672,14128116,7759092,-3326280,-937860,364476,50568,-16577,-1143,264,9,-1).
The conjecture and signatures also apply to the first n distinct Lucas numbers (A000032).

			a(2) = 1*2*3 + 1*2*5 + 1*3*5 + 2*3*5 = 61.

Cf. A000032, A000045, A203245.

z = 50; w[i_] := Fibonacci[i];
t[n_] := Table[w[i], {i, 2, n}]
v[n_] := SymmetricPolynomial[3, t[n]]
tt = Table[v[n], {n, 4, 25}]

a(n) = sum of F(i(1))*F(i(2))*F(i(3)) over all indices i(1)

Deleted program and link based on a conjecture. - N. J. A. Sloane, Dec 22 2024

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A203007 (n-1)-st elementary symmetric function of Fibonacci numbers F(2) to F(n).

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