A173128
a(n) = cosh(2*n*arcsinh(n)).
Original entry on oeis.org
1, 3, 161, 27379, 9478657, 5517751251, 4841332221601, 5964153172084899, 9814664424981012481, 20791777842234580902499, 55106605639755476546020001, 178627672869645203363556318483, 695165908550906808156689590141441
Offset: 0
Cf.
A058331,
A001079,
A037270,
A071253,
A108741,
A132592,
A146311,
A146312,
A146313,
A173115,
A173116,
A173121,
A173127,
A173129,
A173174.
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seq(expand( (1/2)*((n + sqrt(n^2 + 1))^(2*n) + (n - sqrt(n^2 + 1))^(2*n))), n=0..30); # Robert Israel, Apr 05 2016
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Round[Table[Cosh[2 n ArcSinh[n]], {n, 0, 20}]] (* Artur Jasinski *)
Round[Table[1/2 (x - Sqrt[1 + x^2])^(2 x) + 1/2 (x + Sqrt[1 + x^2])^(2 x), {x, 0, 20}]] (* Artur Jasinski, Feb 14 2010 *)
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{a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n^2+1)^(n-k)*n^(2*k))} \\ Seiichi Manyama, Dec 27 2018
-
{a(n) = polchebyshev(n, 1, 2*n^2+1)} \\ Seiichi Manyama, Dec 29 2018
A322790
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Sum_{j=0..k} binomial(2*k,2*j)*(n+1)^(k-j)*n^j.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 17, 5, 1, 1, 99, 49, 7, 1, 1, 577, 485, 97, 9, 1, 1, 3363, 4801, 1351, 161, 11, 1, 1, 19601, 47525, 18817, 2889, 241, 13, 1, 1, 114243, 470449, 262087, 51841, 5291, 337, 15, 1, 1, 665857, 4656965, 3650401, 930249, 116161, 8749, 449, 17, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 3, 17, 99, 577, 3363, 19601, ...
1, 5, 49, 485, 4801, 47525, 470449, ...
1, 7, 97, 1351, 18817, 262087, 3650401, ...
1, 9, 161, 2889, 51841, 930249, 16692641, ...
1, 11, 241, 5291, 116161, 2550251, 55989361, ...
1, 13, 337, 8749, 227137, 5896813, 153090001, ...
-
A[0, k_] := 1; A[n_, k_] := Sum[Binomial[2 k, 2 j]*(n + 1)^(k - j)*n^j, {j, 0, k}]; Table[A[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 26 2018 *)
A322746
a(n) = 1/2 * (-1 + Sum_{k=0..n} binomial(2*n,2*k)*(n+1)^(n-k)*n^k).
Original entry on oeis.org
0, 1, 24, 675, 25920, 1275125, 76545000, 5425069447, 443365544448, 41047124680809, 4245890890571000, 485307363135371051, 60742714406414040000, 8262695239025750162653, 1213734518568509516047560, 191478489107270936785743375, 32288451913272713227175006208
Offset: 0
(sqrt(3) + sqrt(2))^2 = 5 + 2*sqrt(6) = sqrt(25) + sqrt(24). So a(2) = 24.
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{a(n) = 1/2*(-1+sum(k=0, n, binomial(2*n,2*k)*(n+1)^(n-k)*n^k))}
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{a(n) = (polchebyshev(n, 1, 2*n+1)-1)/2}
A173194
a(n) = -sin^2 (2*n*arccos n) = - sin^2 (2*n*arcsin n).
Original entry on oeis.org
0, 0, 9408, 384199200, 54471499791360, 20405558846592060000, 16793517249722147195701440, 26730228454204365035835498694848, 75019085697452515216001640927169855488, 346154755746154620929434271983392498083891520
Offset: 0
Cf.
A132592,
A146311,
A146312,
A146313,
A173115,
A173116 A173121,
A173127,
A173128,
A173129,
A173130,
A173131,
A173133,
A173134,
A173148,
A173151,
A173170,
A173171,
A173174,
A173175,
A173176.
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A173194 := proc(n) ((n+sqrt(n^2-1))^(2*n)-(n-sqrt(n^2-1))^(2*n))^2 ; expand(%/4) ; simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011
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Round[Table[ -N[Sin[2 n ArcSin[n]], 100]^2, {n, 0, 15}]] (* Artur Jasinski *)
Table[FullSimplify[(-1/2 (x - Sqrt[ -1 + x^2])^(2 x) + 1/2 (x + Sqrt[ -1 + x^2])^(2 x))^2], {x, 0, 7}] (* Artur Jasinski, Feb 17 2010 *)
Table[(n^2-1)*ChebyshevU[2*n-1, n]^2, {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *)
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{a(n) = (n^2-1)*n^2*(sum(k=0, n-1, binomial(2*n, 2*k+1)*(n^2-1)^(n-1-k)*n^(2*k)))^2} \\ Seiichi Manyama, Jan 05 2019
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{a(n) = (n^2-1)*polchebyshev(2*n-1, 2, n)^2} \\ Seiichi Manyama, Jan 05 2019
Showing 1-4 of 4 results.