A370260 -id:A370260 - cp's OEIS frontend

A342205 a(n) = T(n,n+1) where T(n,x) is a Chebyshev polynomial of the first kind.

1, 2, 17, 244, 4801, 120126, 3650401, 130576328, 5374978561, 250283080090, 13007560326001, 746411226303612, 46873096812360001, 3197490648645613334, 235451028081583642049, 18614381236112230383376, 1572584048032918633353217

Offset: 0

Seiichi Manyama, Mar 05 2021

Seiichi Manyama, Table of n, a(n) for n = 0..351
Wikipedia, Chebyshev polynomials.

Cf. A115066, A323117, A342206, A342207, A370259, A370260, A370261.

Table[ChebyshevT[n, n + 1], {n, 0, 16}] (* Amiram Eldar, Mar 05 2021 *)

PARI
```
a(n) = polchebyshev(n, 1, n+1);
```
PARI
```
a(n) = round(cos(n*acos(n+1)));
```

a(n) = if(n==0, 1, n*sum(k=0, n, (2*n)^k*binomial(n+k, 2*k)/(n+k)));

a(n) = cos(n*arccos(n+1)).
a(n) = n * Sum_{k = 0..n} (2*n)^k * binomial(n+k,2*k)/(n+k) for n > 0.
From Peter Bala, Mar 11 2024: (Start)
a(2*n+1) == 1 (mod (2*n + 1)^3); a(2*n) == 1 (mod (n + 1)*(2*n)^3).
a(n) = hypergeom([n, -n], [1/2], -n/2). (End)
a(n) ~ exp(1) * 2^(n-1) * n^n. - Vaclav Kotesovec, Mar 12 2024

A370259 a(n) = (T(n,n+1) - 1)/n^3 for n >= 1, where T(n,x) is the n-th Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 2, 9, 75, 961, 16900, 380689, 10498005, 343323841, 13007560326, 560789801881, 27125634729375, 1455389462287489, 85805768251305992, 5515372218107327521, 383931652351786775721, 28778117694539885440129, 2311202255914842794592010, 198009919900727928789497641, 18027589454633803742596931571

Offset: 1

Peter Bala, Mar 11 2024

It appears that a(2*n+1) is always a square, while a(2*n) = (n + 1) * a square. See A370260 and A370261.

Paolo Xausa, Table of n, a(n) for n = 1..350

Cf. A008310, A342205, A370260, A370261.

seq( simplify( (ChebyshevT(n, n+1) - 1)/n^3 ), n = 1..20);

Array[(ChebyshevT[#, #+1]-1)/#^3 &, 20] (* Paolo Xausa, Mar 14 2024 *)

from sympy import chebyshevt
def A370259(n): return (chebyshevt(n,n+1)-1)//n**3 # Chai Wah Wu, Mar 13 2024

a(n) = Sum_{k = 1..n} (2^k)*n^(k-2)*binomial(n+k, 2*k)/(n + k) (shows that a(n) is an integer).
a(n) = (cos(n*arccos(n+1)) - 1)/n^3.
a(n) = (A342205(n) - 1)/n^3.
a(n) = ( (n + 1 + sqrt(n*(n+2)))^n + (n + 1 - sqrt(n*(n+2)))^n - 2 )/(2*n^3).

A370261 a(n) = sqrt(A370259(2*n)/(n+1)) for n >= 1.

Original entry on oeis.org

1, 5, 65, 1449, 46561, 1968525, 103565057, 6531391313, 480749649601, 40482981221781, 3840053099665729, 405275779792031225, 47113209228513626017, 5982545638922153790749, 823992221632687352744961, 122360935410018418223907489, 19489013519781051891806113153

Offset: 1

Peter Bala, Mar 11 2024

The sequence is conjectured to be integral.

Paolo Xausa, Table of n, a(n) for n = 1..300

Cf. A008310, A342205, A370259, A370260.

A370259 := n -> simplify( (ChebyshevT(n, n+1) - 1)/n^3 ):
seq(sqrt(A370259(2*n)/(n+1)), n = 1..20);

Table[Sqrt[(ChebyshevT[2*n, 2*n + 1] - 1)/(2*n)^3/(n + 1)], {n, 20}] (* Paolo Xausa, Jul 24 2024 *)

from math import isqrt
from sympy import chebyshevt
def A370261(n): return isqrt((chebyshevt((m:=n<<1),m+1)-1)//((n+1)*m**3)) # Chai Wah Wu, Mar 13 2024

a(n) = sqrt( (T(2*n, 2*n+1) - 1)/((n+1)*(2*n)^3) ), where T(n, x) is the n-th Chebyshev polynomial of the first kind.

A370262 Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2k + 1) (2*n + 1)^k.

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 1, 14, 49, 49, 1, 30, 243, 729, 729, 1, 55, 847, 5324, 14641, 14641, 1, 91, 2366, 26364, 142805, 371293, 371293, 1, 140, 5670, 101250, 928125, 4556250, 11390625, 11390625, 1, 204, 12138, 324258, 4593655, 36916282, 168962983, 410338673, 410338673

Offset: 0

Peter Bala, Mar 12 2024

The table entries are integers since a(n, k) := binomial(n+k, n-k)/(2*k + 1) * (2*n + 1) gives the entries of the transpose of triangle A082985.

			Triangle begins
 n\k | 0    1      2       3        4        5        6
 - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   0 | 1
   1 | 1    1
   2 | 1    5      5
   3 | 1   14     49      49
   4 | 1   30    243     729      729
   5 | 1   55    847    5324    14641    14641
   6 | 1   91   2366   26364   142805   371293   371293
  ...

A371697 (row sums), A052750 (main diagonal and subdiagonal), A000330 (column 1).

Cf. A008310, A082985, A258708, A370259, A370260.

seq(seq(binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k, k = 0..n), n = 0..10);

Table[Binomial[n + k, n - k] / (2*k + 1) * (2*n + 1)^k, {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 17 2024 *)

n-th row polynomial R(n, x) = Sum_{k = 0..n} T(n, k)*x^k = sqrt( 2* Sum_{k = 0..2*n} (2*n + 1)^(k-1) *binomial(2*n+k+2, 2*k+2)/(2*n + k + 2) * x^k ).
R(n, x)^2 = 2/(x*(2*n + 1)^3) * ( ChebyshevT(2*n+1, 1 + (2*n+1)*x/2) - 1 ).
R(n, 2) = A370260(n).

cp's OEIS Frontend

A342205 a(n) = T(n,n+1) where T(n,x) is a Chebyshev polynomial of the first kind.

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A370259 a(n) = (T(n,n+1) - 1)/n^3 for n >= 1, where T(n,x) is the n-th Chebyshev polynomial of the first kind.

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A370261 a(n) = sqrt(A370259(2*n)/(n+1)) for n >= 1.

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A370262 Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k.

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A370262 Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2k + 1) (2*n + 1)^k.