A118417 -id:A118417 - cp's OEIS frontend

A014480 Expansion of g.f. (1+2x)/(1-2x)^2.

1, 6, 20, 56, 144, 352, 832, 1920, 4352, 9728, 21504, 47104, 102400, 221184, 475136, 1015808, 2162688, 4587520, 9699328, 20447232, 42991616, 90177536, 188743680, 394264576, 822083584, 1711276032, 3556769792, 7381975040, 15300820992, 31675383808, 65498251264

Offset: 0

N. J. A. Sloane

Number of binary trees of size n and height n-1, computed from size n=3 onward; i.e. A014480(n) = A073345(n+3,n+2). (For sizes n=0 through 2 there are no such trees.)
Also determinant of the n X n matrix M(i,j)=binomial(2i+2j,i+j). - Benoit Cloitre, Mar 27 2004
Subdiagonal in triangle displayed in A128196. - Peter Luschny, Feb 26 2007
From Jaume Oliver Lafont, Nov 08 2009: (Start)
From two BBP-type formulas by Knuth, (page 6 of the reference)
Sum_{n>=0} 1/a(n) = 2^(1/2)*log(1+2^(1/2))
Sum_{n>=0} (-1)^n/a(n) = 2^(1/2)*atan(1/2^(1/2))
(End)
Create a triangle with first column T(n,1)=1+4*n for n=0 1 2... The remaining terms T(r,c)=T(r,c-1)+T(r-1,c-1).  T(n,n+1)=a(n). - J. M. Bergot, Dec 18 2012

			(1 + 2*x)/(1-2*x)^2 = 1 + 6*x + 20*x^2 + 56*x^3 + 144*x^4 + 352*x^5 + 832*x^6 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
David Bailey, Peter Borwein, and Simon Plouffe, On the rapid computation of various polylogarithmic constants, in: L. Berggren, J. Borwein, and P. Borwein (eds.), Pi: A Source Book, Springer, New York, NY, 2000.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A054582, A118417, A128196, A132775, A134233.

Leftmost column of A167580 (shifted).

a014480 n = a014480_list !! n
a014480_list = 1 : 6 : map (* 4)
   (zipWith (-) (tail a014480_list) a014480_list)
-- Reinhard Zumkeller, Jan 22 2012

[2^n*(2*n + 1): n in [0..35]]; // Vincenzo Librandi, Oct 20 2014

a:=n-> sum(2^n*n^binomial(j,n)/2,j=1..n): seq(a(n),n=1..29); # Zerinvary Lajos, Apr 18 2009

CoefficientList[ Series[(1 + 2*x)/(1 - 2*x)^2, {x, 0, 28}], x]
LinearRecurrence[{4, -4}, {1, 6}, 29] (* Robert G. Wilson v, Dec 26 2012 *)
Table[2^n (2*n + 1), {n, 0, 28}] (* Fred Daniel Kline, Oct 20 2014 *)

Vec((1+2*x)/(1-2*x)^2+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

a(n) = (2n+1)*2^n = 4a(n-1)-4a(n-2) = 4*A052951(n-1) = a(n-1)+A052951(n) = a(n-1)*(2+4/(2n-1)) = A054582(n, n). - Henry Bottomley, May 16 2001
E.g.f.: x*cosh(sqrt(2)*x) = x + 6x^3/3! + 20x^5/5! + 56x^7/7! +... - Ralf Stephan, Mar 03 2005
From Reinhard Zumkeller, Apr 27 2006: (Start)
a(n) = A118416(n+1,n+1) = A118413(n+1,n+1);
A001511(a(n)) = A003602(a(n));
A117303(a(n)) = a(n). (End)
Row sums of triangle A132775 - Gary W. Adamson, Aug 29 2007
Row sums of triangle A134233 - Gary W. Adamson, Oct 14 2007
From Johannes W. Meijer, Nov 23 2009: (Start)
a(n) = 3*a(n-1) - 2^(n-1)*(2*n-5) with a(0) = 1.
a(n) = 3*a(n-1) - 2*a(n-2) + 2^n with a(0) = 1 and a(1) = 6.
(End)
G.f.: -G(0) where G(k) =  1 - (2*k+2)/(1 - x/(x - (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
E.g.f.: Q(0), where Q(k)= 1 + 4*x/( 1 - 1/(1 + 2*(k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 11 2013

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).

A196525 Decimal expansion of log(1+sqrt(2))/sqrt(2).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 03 2011

			0.6232252401402305133940200802505680... = A091648/A002193.
From _Peter Bala_, Dec 01 2021: (Start)
With N = 10000, the truncated series Sum_{k = 0..N/4 - 1} (-1)^k/((4*k + 1)*(4*k+3)) = 0.6232252[3]014023[16]1339[3659]080... to 27 decimal places. The square bracketed numbers show where this decimal expansion differs from that of (1/sqrt(2))*log(1+sqrt(2)) = 0.6232252(4)014023(05) 1339(4020)080.... The numbers 1, -11, 361 must be added to the square bracketed numbers to give the correct decimal expansion to 27 decimal places. (End)

G. C. Greubel, Table of n, a(n) for n = 0..10000
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, Table section 2.2, L(m=8, r=2, s=1).
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (2.2.3)
Index entries for transcendental numbers.

Cf. A000464, A002193, A001539, A091648, A181048, A181049.

SetDefaultRealField(RealField(100)); Log(Sqrt(2)+1)/Sqrt(2); // G. C. Greubel, Oct 05 2018

RealDigits[Log[1+Sqrt[2]]/Sqrt[2],10,120][[1]] (* Harvey P. Dale, Dec 27 2011 *)
RealDigits[Sum[1/((2 n - 1) 2^n), {n, 1, Infinity}], 10, 120][[1]] (* Fred Daniel Kline, May 23 2019 *)

log(sqrt(2)+1)/sqrt(2) \\ Michel Marcus, Sep 27 2017

Equals Sum_{n>=1} A091337(n)/n = 1 - 1/3 - 1/5 + 1/7 + 1/9 - 1/11 - ...
Equals 2*Sum_{n>=1} (-1)^n/A001539(n). - Michel Marcus, Sep 27 2017
From Fred Daniel Kline, May 23 2019: (Start)
Equals arcsinh(1)/sqrt(2).
Equals Sum_{n>=1} 1/A118417(n-1) = Sum_{n>=1} 1/((2*n - 1)*2^n). (End)
From Peter Bala, Nov 01 2019: (Start)
Equals (1/sqrt(2))*arccoth(sqrt(2)).
Equals 1 - 8*Sum_{n >= 0} (-1)^(n+1)*n/(16*n^2 - 1).
Equals 1 - Integral_{x = 0..inf} exp(-2*x)*cosh(x)/cosh(2*x) dx.
Equals 2*Integral_{x = 0..inf} exp(x)*(exp(2*x) + 1)*(exp(4*x) - 1)/(exp(4*x) + 1)^2 dx - 1. (End)
From Amiram Eldar, Aug 16 2020: (Start)
Equals Sum_{k>=0} (-1)^k * (2*k)!!/(2*k+1)!!.
Equals Integral_{x=0..Pi/4} 1/(cos(x) + sin(x)) dx. (End)
From Peter Bala, Dec 01 2021: (Start)
Equals 2*Sum_{k >= 0} (-1)^k/((4*k + 1)*(4*k + 3)).
Let N be a positive integer divisible by 4. We have the asymptotic expansion (1/sqrt(2))*log(1 + sqrt(2)) - 2*Sum_{k = 0..N/4 - 1} (-1)^k/((4*k + 1)*(4*k + 3)) ~ 1/N^2 - 11/N^4 + 361/N^6 - 24611/N^8 + ..., where the sequence of unsigned coefficients [1, 11, 361, 24611, ...] is A000464. See A181048 and A181049. An example is given below. (End)
Equals 1/Product_{p prime} (1 - Kronecker(8,p)/p), where Kronecker(8,p) = 0 if p = 2, 1 if p == 1 or 7 (mod 8) or -1 if p == 3 or 5 (mod 8). - Amiram Eldar, Dec 17 2023
Equals integral_{x=0..Pi/2} sin^2(x)/(sin(x)+cos(x)) dx [Nahin]. - R. J. Mathar, May 16 2024

A143126 a(n) = (1-2n)*2^n.

Original entry on oeis.org

1, -2, -12, -40, -112, -288, -704, -1664, -3840, -8704, -19456, -43008, -94208, -204800, -442368, -950272, -2031616, -4325376, -9175040, -19398656, -40894464, -85983232, -180355072, -377487360, -788529152, -1644167168, -3422552064, -7113539584, -14763950080

Offset: 0

Paul Barry, Jul 26 2008

Hankel transform of abs(A002420) (which is 2*0^n - binomial(2n,n)/(2n-1)).

Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A002420, A118417, A121314.

a[n_] := (1-2n)*2^n; Array[a, 40, 0] (* Amiram Eldar, Oct 01 2022 *)

G.f.: (1-6x)/(1-2x)^2;
a(n) = Sum_{k=0..n} A121314(n,k)*(-1)^k*2^(3n-2k). - Philippe Deléham, Oct 31 2008
From Amiram Eldar, Oct 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 - arcsinh(1)/sqrt(2).
Sum_{n>=0} (-1)^n/a(n) = 1 + arctan(1/sqrt(2))/sqrt(2). (End)

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A014480 Expansion of g.f. (1+2*x)/(1-2*x)^2.

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

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A196525 Decimal expansion of log(1+sqrt(2))/sqrt(2).

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A143126 a(n) = (1-2n)*2^n.

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A014480 Expansion of g.f. (1+2x)/(1-2x)^2.

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