A195727 -id:A195727 - cp's OEIS frontend

A050166 Triangle T(n,k) = M(2n,k,-1), with 0 <= k <= n, n >= 0, and array M is defined in A050144.

1, 1, 2, 1, 4, 5, 1, 6, 14, 14, 1, 8, 27, 48, 42, 1, 10, 44, 110, 165, 132, 1, 12, 65, 208, 429, 572, 429, 1, 14, 90, 350, 910, 1638, 2002, 1430, 1, 16, 119, 544, 1700, 3808, 6188, 7072, 4862, 1, 18, 152, 798, 2907, 7752, 15504, 23256, 25194, 16796

Offset: 0

Clark Kimberling

Sometimes called Catalan's triangle, although this term is usually reserved for several other triangles!
T is a mirror image of the array in A039598.
Given (1) = row 0, then the sum of terms with alternating signs in row r of A050166 = (-1)^r * A000108(n); where A000108 = 1, 1, 2, 5, 14, 42, ...the Catalan numbers. - Herb Conn
The diagonals of this triangle are self-convolutions of the main diagonal A000108(n+1): 1, 2, 5, 14, 42, 132, 429, ... - Philippe Deléham, May 25 2005
The multiplicities of the eigenvalues of the middle cubes are related to this triangle. The middle cube in Q_3 has eigenvalues -2, -1, 1, 2 with multiplicities 1, 2, 2, 1. The middle cube in Q_5 has eigenvalues -3, -2, -1, 1, 2, 3 with multiplicities 1, 4, 5, 5, 4, 1. The middle cube in Q_7 has eigenvalues -4, -3, -2, -1, 1, 2, 3, 4 with multiplicities 1, 6, 14, 14, 14, 14, 6, 1, etc. - Ke Qiu, Apr 05 2019

			Triangle begins:
  1;
  1,  2;
  1,  4,  5;
  1,  6, 14, 14;
  1,  8, 27, 48, 42;
  ...

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
Y. Jiang, K. Qiu, R. Qiu, and J. Shen, On the spectrum of the middle-cube, Congressus Numerantium, 195 (2009), 195-204.
A. Nkwanta, Lattice paths and RNA secondary structures, in: Nathaniel Dean, African Americans in Mathematics, AMS and DIMACS, 1997, ISBN 978-0-8218-0678-4, pp. 137-147.

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 29.
E. H. M. Brietzke, An identity of Andrews and a new method for the Riordan array proof of combinatorial identities, Discrete Math., 308 (2008), 4246-4262.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
L. W. Shapiro, W.-J. Woan and S. Getu, Runs, slides and moments, SIAM J. Alg. Discrete Methods, 4 (1983), 459-466.

Mirror image of A039598.

Flat(List([0..10], n-> List([0..n], k-> 2*(n-k+1)* Binomial(2*n+1, k)/(2*n-k+2) ))); # G. C. Greubel, Apr 05 2019

[[2*(n-k+1)*Binomial(2*n+1,k)/(2*n-k+2): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 05 2019

Table[2*Binomial[2n+1, k]*(n-k+1)/(2*n-k+2), {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 05 2019 *)

{T(n,k) = 2*(n-k+1)*binomial(2*n+1,k)/(2*n-k+2)}; \\ G. C. Greubel, Apr 05 2019

[[2*(n-k+1)*binomial(2*n+1,k)/(2*n-k+2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 05 2019

From Henry Bottomley, Sep 24 2001: (Start)
T(n, k) = C(2n+1, k)*2*(n-k+1)/(2n-k+2) = A039598(n, n-k)
T(n, k) = T(n-1, k) + 2*T(n-1, k-1) + T(n-1, k-2), with T(0, 0) = 1 and T(n, k) = 0 if n < 0 or n < k. (End)
Sum_{0<=k<=n} T(n,k)*x^k = A000012(n), A001700(n), A194723(n+1), A194724(n+1), A194725(n+1), A194726(n+1), A195727(n+1), A194728(n+1), A194729(n+1), A194730(n+1) for x = 0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Nov 03 2011

More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 2001

A295257 Expansion of e.g.f. cot(x)(1 - sqrt(1 - 4tan(x)))/2.

Original entry on oeis.org

1, 1, 4, 32, 368, 5656, 109024, 2533712, 68995328, 2155513216, 76014982144, 2987332904192, 129473128921088, 6135478762187776, 315609465774936064, 17515027337549545472, 1043104219010147483648, 66358462250378681614336, 4491141928841064201846784, 322219449242531127348887552

Offset: 0

Ilya Gutkovskiy, Nov 18 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..365

Cf. A000108, A195727, A295237, A295254, A295255, A295256, A295258.

S:= series(cot(x)*(1 - sqrt(1 - 4*tan(x)))/2, x, 32):
seq(n!*coeff(S,x,n),n=0..30); # Robert Israel, Nov 18 2017

nmax = 19; CoefficientList[Series[Cot[x] (1 - Sqrt[1 - 4 Tan[x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-Tan[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: 1/(1 - tan(x)/(1 - tan(x)/(1 - tan(x)/(1 - tan(x)/(1 - ...))))), a continued fraction.

a(n) ~ sqrt(17/2) * n^(n-1) / (exp(n) * (arctan(1/4))^(n-1/2)). - Vaclav Kotesovec, Nov 18 2017

A195628 Decimal expansion of arctan(4).

Original entry on oeis.org

1, 3, 2, 5, 8, 1, 7, 6, 6, 3, 6, 6, 8, 0, 3, 2, 4, 6, 5, 0, 5, 9, 2, 3, 9, 2, 1, 0, 4, 2, 8, 4, 7, 5, 6, 3, 1, 1, 8, 4, 4, 4, 0, 6, 0, 1, 3, 0, 6, 3, 6, 8, 8, 4, 3, 3, 6, 0, 0, 9, 6, 3, 8, 1, 4, 8, 6, 5, 5, 3, 0, 8, 3, 5, 5, 5, 4, 6, 2, 4, 0, 2, 7, 3, 9, 4, 8, 3, 2, 5, 5, 0, 0, 2, 3, 5, 6, 5, 4, 2

Offset: 1

Clark Kimberling, Sep 23 2011

			arctan(4) = 1.3258176636680324650592392104284756311844406013...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A195727, A195731, A195621.

SetDefaultRealField(RealField(125)); Arctan(4); // G. C. Greubel, Apr 13 2023

r = 4;
N[ArcTan[r], 100]
RealDigits[%]  (* A195628 *)
N[ArcCot[r], 100]
RealDigits[%]  (* A195727 *)
N[ArcSec[r], 100]
RealDigits[%]  (* A195731 *)
N[ArcCsc[r], 100]
RealDigits[%]  (* A195621 *)

atan(4) \\ Charles R Greathouse IV, Nov 20 2024

numerical_approx( arctan(4), digits=125) # G. C. Greubel, Apr 13 2023

Equals arcsin(4/sqrt(17)) = arccos(1/sqrt(17)). - Amiram Eldar, Jul 11 2023

A344906 Decimal expansion of Sum_{k>=0} arctan(1/2^k).

Original entry on oeis.org

1, 7, 4, 3, 2, 8, 6, 6, 2, 0, 4, 7, 2, 3, 4, 0, 0, 0, 3, 5, 0, 4, 3, 3, 7, 6, 5, 6, 1, 3, 6, 4, 1, 6, 2, 8, 5, 8, 1, 3, 8, 3, 1, 1, 8, 5, 4, 2, 8, 2, 0, 6, 5, 2, 3, 0, 0, 4, 5, 6, 9, 5, 7, 2, 0, 5, 6, 5, 5, 1, 7, 6, 5, 2, 2, 7, 4, 9, 2, 0, 5, 5, 8, 1, 6, 5, 8, 6, 8

Offset: 1

Daniel Hoyt, Jun 01 2021

This number can be interpreted geometrically as the angle in radians of a fan made of stacked right triangles, with the length to height ratio doubling each successive triangle as seen in the illustration.

Since this angle exceeds Pi/2, the set of rotation angles used in the CORDIC algorithm covers an angle range sufficient to compute sine and cosine for any angle between 0 and Pi/2. This means the algorithm can converge to any angle in that range through appropriate combinations of these basic rotations. - Daniel Hoyt, Oct 25 2024

			1.743286620472340003...

Daniel Hoyt, Illustration of this angle's arctan relationship.
Wikipedia, CORDIC.

Cf. A003881, A065445, A073000, A195727, A195782.

Digits:= 140:
evalf(sum(arccot(2^k), k=0..infinity));  # Alois P. Heinz, Jun 02 2021

PARI
```
suminf(k=0, atan(1/2^k))
```

sumalt(k=1, ((-1)^(k+1))*2^(2*k-1)/((2^(2*k-1)-1)*(2*k-1)))

Equals Sum_{k>=1} (-1)^(k+1)*2^(2*k-1)/((2^(2*k-1)-1)*(2*k-1)).

cp's OEIS Frontend

A050166 Triangle T(n,k) = M(2n,k,-1), with 0 <= k <= n, n >= 0, and array M is defined in A050144.

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A295257 Expansion of e.g.f. cot(x)*(1 - sqrt(1 - 4*tan(x)))/2.

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A195628 Decimal expansion of arctan(4).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A344906 Decimal expansion of Sum_{k>=0} arctan(1/2^k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

PARI

Formula

A295257 Expansion of e.g.f. cot(x)(1 - sqrt(1 - 4tan(x)))/2.