A246645 -id:A246645 - cp's OEIS frontend

A168467 a(n) = Product_{k=0..n} ((2k+2)(2*k+3))^(n-k).

1, 6, 720, 3628800, 1316818944000, 52563198423859200000, 327312129899898454671360000000, 428017682605583614976547335700480000000000, 152240508705590071980086429193304853792686080000000000000

Offset: 0

Paul Barry, Nov 26 2009

Hankel transform of A000698(n+1).
The sequence 1,1,6,720,... with general term Product_{k=0..n, ((2k+1)(2k+0^k))^(n-k)} is the Hankel transform of A112934. - Paul Barry, Dec 04 2009
a(n) is also the determinant of the n X n matrix M(i,j) = i^(2*j)*sinh(2*j*arccsch(i))/(2*sqrt(i^2+1)), with i and j from 1 to n, which is the same matrix generated by sequences of length n by the linear recurrences with kernel { 2*(k^2 + z), -k^4 }, and initial conditions { 1, 2*(k^2 + z) }, with k from 1 to n, and z = 2. Regardless of the value of z, for every n, the determinant of the n X n matrix of polynomials generated gives always a(n) as result. - Federico Provvedi, Feb 01 2021

			From _Federico Provvedi_, Apr 01 2021: (Start)
From both formulas in the comment above and in particular with z=2 from the linear recurrences, the determinant of the 5 X 5 matrix: ( (1,6,35,204,1189), (1,12,128,1344,14080),(1,22,403,7084,123205), (1,36,1040,28224,749824), (1,54,2291,89964,3426181) ) = 1316818944000 = a(5).
For a generic z, the determinant doesn't change as shown in this example, where the determinant of the 3 X 3 square matrix:
( ( 1, 2*(z+1), (2*z + 1)*(2*z+3)  ),
  ( 1, 2*(z+4), 4*(z+6)*(z+2)      ),
  ( 1, 2*(z+9), (2*z + 9)(2*z + 27)) ) = 720 = a(3). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..28

Cf. A000178, A000698, A098694, A112934, A306635, A001109, A099157, A246645, A202768, A000142.

Table[2^(n^2 + 2*n + 23/24) Glaisher^(3/2) Pi^(-n/2 - 3/4) BarnesG[n + 2] BarnesG[n + 5/2]/E^(1/8), {n, 0, 10}] (* Vladimir Reshetnikov, Sep 06 2016 *)
Table[Product[((2k+2)(2k+3))^(n-k),{k,0,n}],{n,0,10}] (* Harvey P. Dale, Dec 26 2019 *)
Table[Det@Table[LinearRecurrence[{2*k^2,-k^4},{1, 2*k^2},n], {k, 1, n}], {n,1,20}] (* Federico Provvedi, Feb 01 2021 *)
Det@Expand@Array[(#1^(2 #2))/(4 Sqrt[1 + #1^2])((Sqrt[1+1/#1^2]+1/#1)^(2 #2)-(Sqrt[1+1/#1^2]-1/#1)^(2 #2))&,{#,#}]&/@Range[20] (* Federico Provvedi, Apr 01 2021 *)

from math import prod
def A168467(n): return prod(((m:=k+1<<1)*(m+1))**(n-k) for k in range(1,n+1))*3**n<Chai Wah Wu, Nov 26 2023

G.f.: Q(0)/(2*x) -1/x, where Q(k) = 1  + 1/(1 -(2*k+1)!*x/((2*k+1)!*x + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 17 2013
a(n) = Product_{k=1..n} (2*k+1)!. - Vladimir Reshetnikov, Sep 06 2016
a(n) ~ A^(-1/2) * 2^(n^2 + 3*n + 53/24) * exp((-3/2)*n^2 + (-5/2)*n + 1/24) * n^(n^2 + (5/2)*n + 35/24) * Pi^((n+1)/2), where A = A074962 is the Glaisher-Kinkelin constant. - Vladimir Reshetnikov, Sep 06 2016
a(n) = A000178(2*n + 1) / A098694(n). - Vaclav Kotesovec, Oct 28 2017
a(n) = A202768(n)*A000142(n). - Federico Provvedi, Feb 01 2021
For n > 0, a(n) = n * (2*n+1) * sqrt(BarnesG(2*n)) * Gamma(2*n)^2 / (sqrt(Gamma(n)) * 2^((n-3)/2)). - Vaclav Kotesovec, Nov 27 2024

A247512 The curvature (rounded down) of touching circles inscribed in a special way in the smaller segment of circle of radius 10/9 divided by a chord of length 4/3.

Original entry on oeis.org

9, 10, 13, 20, 35, 64, 119, 224, 428, 821, 1576, 3030, 5828, 11215, 21584, 41545, 79968, 153931, 296306, 570371, 1097933, 2113463, 4068308, 7831289, 15074840, 29018319, 55858826, 107525476, 206981225, 398428629, 766955420, 1476351286, 2841903278, 5470523390

Offset: 0

Kival Ngaokrajang, Sep 18 2014

Refer to comment of A240926. This is the companion of A247335. After the first two terms, the curvatures seem to be non-integer.

The actual rational curvatures can be computed. See part II of the W. Lang link for the proofs of the statements given in the formula section.

			The first curvatures r(n) are 9, 10, 121/9, 1690/81, 25921/729, 420250/6561, 7027801/59049, 119508490/531441,... - _Wolfdieter Lang_, Sep 30 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Wolfdieter Lang, Curvature computation for A247335 and A247512.
Index entries for sequences related to Chebyshev polynomials.

Cf. A240926, A247335.

Cf. A246643, A049310, A246645. - Wolfdieter Lang, Sep 30 2014

r[0] := 9; r[n_] := r[n] = (11*r[n - 1] - 9 + 20*Sqrt[(r[n - 1] - 9)*r[n - 1]/10])/9; Table[Floor[r[n]], {n, 0, 30}] (* G. C. Greubel, Dec 20 2017 *)

{
r=0.1;print1(floor(9/(10*r)),", ");r1=r;
for (n=1,50,
if (n<=1,ab=2-r,ab=sqrt(ac^2+r^2));
ac=sqrt(ab^2-r^2);
if (n<=1,z=0,z=(Pi/2)-atan(ac/r)+asin((r1-r)/(r1+r));r1=r);
b=acos(r/ab)-z;
r=r*(1-cos(b))/(1+cos(b));
print1(floor(9/(10*r)),", ")
)
}

From Wolfdieter Lang, Sep 30 2014 (Start)
a(n) = floor(r(n)) with the rational curvatures r(n) satisfying the one step nonlinear recurrence relation r(n) = (11*r(n-1) - 9 + 20*sqrt((r(n-1) - 9)*r(n-1)/10))/9 with input r(0) = 9. (In the link r(n) is called b'(n).)
r(n) =  A246643(n)/9^(n-1)  = (9/2)*(1 + S(n, 22/9) - (11/9)*S(n-1, 22/9)), n >= 0, with Chebyshev/s S-polynomials (see A049310). 9^n*S(n, 22/9) = A246645(n). See A246643 for more details. (End)

Edited: Keyword easy and Chebyshev index link added. Wolfdieter Lang, Sep 30 2014

A246643 A sequence used in the touching circle problem described in A247512.

Original entry on oeis.org

1, 10, 121, 1690, 25921, 420250, 7027801, 119508490, 2050368961, 35341836010, 610665665401, 10564982353210, 182902930753921, 3167536046903290, 54865571909148121, 950426408617182250, 16464857882672822401, 285238628280432626890, 4941562979309619843961

Offset: 0

Wolfdieter Lang, Sep 30 2014

This sequence appears in the touching circle problem considered in A247512. There the rational curvatures are b'(n) = a(n)/9^(n-1), and A247512(n) = floor(b'(n)).

See the W. Lang link, part II) with the details where B'(n) plays the role of a(n).

G. C. Greubel, Table of n, a(n) for n = 0..800
Wolfdieter Lang, Curvature computation for A247335 and A247512.
Index entries for linear recurrences with constant coefficients, signature (31,-279,729).

Cf. A247512, A246645.

I:=[1, 10, 121]; [n le 3 select I[n] else 31*Self(n-1) - 279*Self(n-2) + 729*Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 20 2017

CoefficientList[Series[(1 - 21*x + 90*x^2)/((1 - 9*x)*(1 - 22*x + 81*x^2)), {x, 0, 50}], x] (* or *) LinearRecurrence[{31,-279,729}, {1, 10, 121}, 50] (* G. C. Greubel, Dec 20 2017 *)

Vec(-(6*x-1)*(15*x-1)/((9*x-1)*(81*x^2-22*x+1)) + O(x^100)) \\ Colin Barker, Sep 30 2014

One step recurrence: a(n) = 11*a(n-1) - 9^(n-1) + 20*sqrt((a(n-1) - 9^(n-1))*a(n-1)/10), a(0) = 1, n >= 1.
a(n) =  (9^n)*(1 + S(n, 22/9) - (11/9)*S(n-1, 22/9))/2, with Chebyshev's S-polynomials (see A049310). For 9^n*S(n, 22/9) see A246645(n). The positive integer sequence sqrt((a(n) - 9^n)*a(n)/10) = A246645(n-1).
O.g.f.: (1 - 21*x + 90*x^2)/((1 - 9*x)*(1 - 22*x + 81*x^2)) = (1/2)*((1 -11*x)/(1 - 22*x + 81*x^2 ) - 1/(1 - 9*x)).
For the proofs see the W. Lang link with a(n) = B'(n).
a(n) = 31*a(n-1)-279*a(n-2)+729*a(n-3). - Colin Barker, Sep 30 2014

A282854 34-gonal numbers: a(n) = n(32n-30)/2.

Original entry on oeis.org

0, 1, 34, 99, 196, 325, 486, 679, 904, 1161, 1450, 1771, 2124, 2509, 2926, 3375, 3856, 4369, 4914, 5491, 6100, 6741, 7414, 8119, 8856, 9625, 10426, 11259, 12124, 13021, 13950, 14911, 15904, 16929, 17986, 19075, 20196, 21349, 22534, 23751

Offset: 0

Daniel Mohebiravesh, Feb 23 2017

Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A246645, A254474, A261191.

Table[n(32n-30)/2, {n,50}]
PolygonalNumber[34,Range[0,40]] (* or *) LinearRecurrence[{3,-3,1},{0,1,34},40] (* The first program requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 26 2018 *)

a(n)=n*(16*n-15) \\ Charles R Greathouse IV, Feb 27 2017

From Nikolaos Pantelidis, Feb 09 2023 : (Start)
G.f.: x*(1 + 31*x)/(1 - x)^3.
E.g.f.: exp(x)*(x + 16*x^2). (End)

cp's OEIS Frontend

A168467 a(n) = Product_{k=0..n} ((2*k+2)*(2*k+3))^(n-k).

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A247512 The curvature (rounded down) of touching circles inscribed in a special way in the smaller segment of circle of radius 10/9 divided by a chord of length 4/3.

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A246643 A sequence used in the touching circle problem described in A247512.

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Author

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Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A282854 34-gonal numbers: a(n) = n*(32*n-30)/2.

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Author

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Programs

Mathematica

PARI

Formula

A168467 a(n) = Product_{k=0..n} ((2k+2)(2*k+3))^(n-k).

A282854 34-gonal numbers: a(n) = n(32n-30)/2.