A118415 -id:A118415 - cp's OEIS frontend

A058962 a(n) = 2^(2n)(2*n+1).

1, 12, 80, 448, 2304, 11264, 53248, 245760, 1114112, 4980736, 22020096, 96468992, 419430400, 1811939328, 7784628224, 33285996544, 141733920768, 601295421440, 2542620639232, 10720238370816, 45079976738816, 189115999977472, 791648371998720

Offset: 0

N. J. A. Sloane, Jan 13 2001

Denominators in expansion of -1/2*i*Pi+i*arcsin((1+1/4*x^2)/(1-1/4*x^2)), where i=sqrt(-1); numerators are all 1.
Bisection of A001787. That is, a(n) = A001787(2n+1). - Graeme McRae, Jul 12 2006
Denominators of odd terms in expansion of 2*arctanh(s/2); numerators are all 1. - Gerry Martens, Jul 26 2015
Reciprocals of coefficients of Taylor series expansion of sinh(x/2) / (x/2). - Tom Copeland, Feb 03 2016

Harry J. Smith, Table of n, a(n) for n = 0..200
G. A. Campbell, Physical theory of the electric wave-filter, Bell Syst. Tech. J., 1 (1922), 1-32, see Eq. (15d). Also reprinted in M. E. Van Valkebburg, ed., Circuit Theory, Dowden, Hutchinson and Ross, 1974.
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Cf. A001787, A002391, A073000, A118413, A118415.
Cf. A154920. - Jaume Oliver Lafont, Jan 29 2009
Factor of the LS1[-2,n] matrix coefficients in A160487. - Johannes W. Meijer, May 24 2009

[2^(2*n)*(2*n+1) : n in [0..30]]; // Wesley Ivan Hurt, Aug 07 2015

a[n_] := 1/SeriesCoefficient[2 ArcTanh[s/2],{s,0,n}]
Table[a[n], {n, 1, 40, 2}] (* Gerry Martens, Jul 26 2015 *)
Table[2^(2 n) (2 n + 1), {n, 0, 40}] (* Vincenzo Librandi, Aug 08 2015 *)
a[ n_] := With[{m = 2 n + 2}, If[ n < 0, -a[-1 - n] 4^(m - 1), m! SeriesCoefficient[ x^2 D[x Sinc[I x]^2, x]/2, {x, 0, m}]]]; (* Michael Somos, Jun 18 2017 *)

A058962(n)=2^(2*n)*(2*n+1) \\ M. F. Hasler, Aug 11 2015

{a(n) = my(m = 2*n + 2); if( n<0, -a(-1 - n) * 4^(m - 1), m! * polcoeff( x^2 * deriv(x * sinc(I*x + x * O(x^m))^2, x) / 2, m))}; /* Michael Somos, Jun 18 2017 */

Central terms of the triangle in A118416: a(n) = A118416(2*n+1, n+1) - Reinhard Zumkeller, Apr 27 2006
Sum_{n>=0} 1/a(n) = log(3). - Jaume Oliver Lafont, May 22 2007; corrected by Jaume Oliver Lafont, Jan 26 2009
a(n) = 4((2n+1)/(2n-1))*a(n-1) = 4*a(n-1)+2^(2n+1) = 8*a(n-1)-16*a(n-2). - Jaume Oliver Lafont, Dec 09 2008
G.f.: (1+4*x)/(1-4*x)^2. - Jaume Oliver Lafont, Jan 29 2009
E.g.f.: exp(4*x)*(1+8*x). - Robert Israel, Aug 10 2015
a(n) = -a(-1-n) * 4^(2*n+1) for all n in Z. - Michael Somos, Jun 18 2017
a(n) = Sum_{k = 0..n} (2*k + 1)^2*binomial(2*n + 1, n - k). - Peter Bala, Feb 25 2019
Sum_{n>=0} (-1)^n/a(n) = 2 * arctan(1/2) = 2 * A073000. - Amiram Eldar, Jul 03 2020

A118413 Triangle read by rows: T(n,k) = (2n-1)2^(k-1), 0

Original entry on oeis.org

1, 3, 6, 5, 10, 20, 7, 14, 28, 56, 9, 18, 36, 72, 144, 11, 22, 44, 88, 176, 352, 13, 26, 52, 104, 208, 416, 832, 15, 30, 60, 120, 240, 480, 960, 1920, 17, 34, 68, 136, 272, 544, 1088, 2176, 4352, 19, 38, 76, 152, 304, 608, 1216, 2432, 4864, 9728, 21, 42, 84, 168

Offset: 1

Reinhard Zumkeller, Apr 27 2006

Central terms give A118415; row sums give A118414;
T(n,1) = A005408(n-1);
T(n,2) = A016825(n-1) for n>1;
T(n,3) = A017113(n-1) for n>2;
T(n,4) = A051062(n-1) for n>3;
T(n,n-2) = A052951(n-1) for n>2;
T(n,n) = A014480(n-1) = A118416(n,n);
A001511(T(n,k)) = A002260(n,k);
A003602(T(n,k)) = A002024(n,k).
G.f.: x*y*(1 + x + 2*x*y - 6*x^2*y)/((1 - x)^2*(1 - 2*x*y)^2). - Stefano Spezia, Dec 22 2024

			   1
   3   6
   5  10  20
   7  14  28  56
   9  18  36  72 144
  11  22  44  88 176 352
  13  26  52 104 208 416  832
  15  30  60 120 240 480  960 1920
  17  34  68 136 272 544 1088 2176 4352
  19  38  76 152 304 608 1216 2432 4864 9728
  ...

Cf. A001511, A002024, A002260, A003602, A005408, A014480, A016825, A017113, A051062, A052951, A117303, A118414, A118415, A118416, A117303.

Select[Flatten[Table[(2n-1)2^(k-1),{n,20},{k,0,n}]],IntegerQ] (* Harvey P. Dale, Jan 17 2024 *)

from math import isqrt, comb
def A118413(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    return ((a<<1)-1)<Chai Wah Wu, Jun 20 2025

A118416 Triangle read by rows: T(n,k) = (2k-1)2^(n-1), 0 < k <= n.

Original entry on oeis.org

1, 2, 6, 4, 12, 20, 8, 24, 40, 56, 16, 48, 80, 112, 144, 32, 96, 160, 224, 288, 352, 64, 192, 320, 448, 576, 704, 832, 128, 384, 640, 896, 1152, 1408, 1664, 1920, 256, 768, 1280, 1792, 2304, 2816, 3328, 3840, 4352, 512, 1536, 2560, 3584, 4608, 5632, 6656, 7680

Offset: 1

Reinhard Zumkeller, Apr 27 2006

Row sums give A014477: Sum_{k=1..n} T(n,k) = A014477(n-1);
central terms give A118415; T(2*k-1,k) = A058962(k-1);
T(n,1) = A000079(n-1);
T(n,2) = A007283(n-1) for n > 1;
T(n,3) = A020714(n-1) for n > 2;
T(n,4) = A005009(n-1) for n > 3;
T(n,5) = A005010(n-1) for n > 4;
T(n,n-1) = A118417(n-1) for n > 1;
T(n,n) = A014480(n-1) = A118413(n,n);
A001511(T(n,k)) = A002024(n,k);
A003602(T(n,k)) = A002260(n,k).
The alternating row sums, Sum_{k=1..n} (-1)^(k+1)*T(n,k), are: (a) in odd rows, the central term, T(n,(n+1)/2) = A058962((n-1)/2); (b) in even rows, the negation of the average of the two central terms, -(T(2n,n) + T(2n,+1))/2 = -A018215(m/2). The absolute values of the alternating row sums give the plain row means, Sum_{k=1..n} T(n,k)/n; the alternating sign row means are (-2)^(n-1). - Gregory Gerard Wojnar, Feb 10 2024

			Triangle begins:
   1;
   2,   6;
   4,  12,  20;
   8,  24,  40,  56;
  16,  48,  80, 112, 144;
  32,  96, 160, 224, 288, 352;
  64, 192, 320, 448, 576, 704, 832;

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Cf. A118413, A117303, A054582.

a118416 n k = a118416_tabl !! (n-1) !! (k-1)
a118416_row 1 = [1]
a118416_row n = (map (* 2) $ a118416_row (n-1)) ++ [a014480 (n-1)]
a118416_tabl = map a118416_row [1..]
-- Reinhard Zumkeller, Jan 22 2012

A118416 := proc(n,k) 2^(n-1)*(2*k-1) ; end proc: # R. J. Mathar, Sep 04 2011

Flatten[Table[(2k-1)2^(n-1),{n,10},{k,n}]] (* Harvey P. Dale, Aug 26 2014 *)

from math import isqrt
def A118416(n): return (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(1-a)+(n<<1)-1<Chai Wah Wu, Jun 20 2025

T(n,k) = 2*T(n-1,k), 1 <= k < n; T(n,n) = A014480(n-1).

cp's OEIS Frontend

A058962 a(n) = 2^(2n)(2*n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A118413 Triangle read by rows: T(n,k) = (2n-1)2^(k-1), 0

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A118416 Triangle read by rows: T(n,k) = (2k-1)2^(n-1), 0 < k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Formula

cp's OEIS Frontend

A058962 a(n) = 2^(2*n)*(2*n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A118413 Triangle read by rows: T(n,k) = (2*n-1)*2^(k-1), 0

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A118416 Triangle read by rows: T(n,k) = (2*k-1)*2^(n-1), 0 < k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Formula

A058962 a(n) = 2^(2n)(2*n+1).

A118413 Triangle read by rows: T(n,k) = (2n-1)2^(k-1), 0

A118416 Triangle read by rows: T(n,k) = (2k-1)2^(n-1), 0 < k <= n.