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A173174 a(n) = cosh(2*n*arcsinh(sqrt(n))). 5
1, 3, 49, 1351, 51841, 2550251, 153090001, 10850138895, 886731088897, 82094249361619, 8491781781142001, 970614726270742103, 121485428812828080001, 16525390478051500325307, 2427469037137019032095121, 382956978214541873571486751, 64576903826545426454350012417, 11591229031806966336496244914595 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..321

Wikipedia, Chebyshev polynomials.

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n+1)^(n-k)*n^k. - Seiichi Manyama, Dec 26 2018

a(n) = T_{n}(2*n+1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018

MAPLE

A173174 := proc(n) cosh(2*n*arcsinh(sqrt(n))) ; expand(%) ; simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011

MATHEMATICA

Table[Round[N[Cosh[(2 n) ArcSinh[Sqrt[n]]], 100]], {n, 0, 30}] (* Artur Jasinski *)

Join[{1}, a[n_]:=Sum[Binomial[2 n, 2 k] (n + 1)^(n - k) n^k, {k, 0, n}]; Array[a, 25]] (* Vincenzo Librandi, Dec 29 2018 *)

PROG

(PARI) {a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n+1)^(n-k)*n^k)} \\ Seiichi Manyama, Dec 26 2018

(PARI) {a(n) = polchebyshev(n, 1, 2*n+1)} \\ Seiichi Manyama, Dec 29 2018

(MAGMA) [&+[Binomial(2*n, 2*k)*(n+1)^(n-k)*n^k: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Dec 29 2018

CROSSREFS

Cf. A132592, A146311 - A146313, A173115, A173116 A173121, A173127 - A173131, A173133, A173134, A173148, A173151, A173170, A173171.

Cf. A322746.

Main diagonal of A322790.

Sequence in context: A208935 A202534 A277497 * A302466 A303248 A187664

Adjacent sequences:  A173171 A173172 A173173 * A173175 A173176 A173177

KEYWORD

nonn

AUTHOR

Artur Jasinski, Feb 11 2010

EXTENSIONS

More terms from Seiichi Manyama, Dec 26 2018

STATUS

approved

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Last modified January 29 04:57 EST 2020. Contains 331335 sequences. (Running on oeis4.)