A118413 -id:A118413 - cp's OEIS frontend

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

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).

A117303 Self-inverse permutation of the natural numbers based on the bijection (2x-1)2^(y-1) <--> (2y-1)2^(x-1).

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 8, 7, 16, 12, 32, 10, 64, 24, 128, 9, 256, 48, 512, 20, 1024, 96, 2048, 14, 4096, 192, 8192, 40, 16384, 384, 32768, 11, 65536, 768, 131072, 80, 262144, 1536, 524288, 28, 1048576, 3072, 2097152, 160, 4194304, 6144, 8388608, 18, 16777216

Offset: 1

Reinhard Zumkeller, Apr 24 2006

nonn

a(a(n)) = n; fixed points A014480: a(A014480(n)) = A014480(n). - Reinhard Zumkeller, Apr 27 2006

Cf. A000265, A007814, A006519, A005408, A000079.

Cf. A118413, A118416.

a:= n-> (j-> (2*j+1)*2^((n/2^j-1)/2))(padic[ordp](n, 2)):
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 23 2019

a[n_] := (2 IntegerExponent[2 n, 2] - 1)*2^((n/2^IntegerExponent[n, 2] + 1)/2 - 1); Array[a, 50] (* Jean-François Alcover, Mar 12 2019 *)

def A117303(n): return (((m:=(n&-n).bit_length())<<1)-1)*(1<<(n>>m)) # Chai Wah Wu, Jul 14 2022

a(n) = (2*A001511(n) - 1) * 2^(A003602(n) - 1).

Spelling corrected by Jason G. Wurtzel, Aug 23 2010

A052951 Expansion of (1 + x - 2x^2)/(1 - 2x)^2.

Original entry on oeis.org

1, 5, 14, 36, 88, 208, 480, 1088, 2432, 5376, 11776, 25600, 55296, 118784, 253952, 540672, 1146880, 2424832, 5111808, 10747904, 22544384, 47185920, 98566144, 205520896, 427819008, 889192448, 1845493760, 3825205248, 7918845952

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Equals binomial transform of A042948 starting with "1": (1, 4, 5, 8, 9, 12, 13, ...) = terms > 0, == 0 or 1 mod 4. - Gary W. Adamson, Feb 07 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
O. Aichholzer, A. Asinowski, and T. Miltzow, Disjoint compatibility graph of non-crossing matchings of points in convex position, arXiv preprint arXiv:1403.5546 [math.CO], 2014.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1021.
Agustín Moreno Cañadas, Hernán Giraldo, Gabriel Bravo Rios, On the Number of Sections in the Auslander-Reiten Quiver of Algebras of Dynkin Type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 8 (2017), pp. 1631-1654.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A042948, A118413.

Concatenation([1], List([1..40], n-> 2^(n-1)*(2*n+3) )); # G. C. Greubel, Oct 21 2019

I:=[1, 5, 14]; [n le 3 select I[n] else 4*Self(n-1)-4*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jun 22 2012

spec:= [S,{S=Prod(Union(Sequence(Union(Z,Z)),Z),Sequence(Union(Z,Z)))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(`if`(n=0, 1, 2^(n-1)*(2*n+3)), n=0..40); # G. C. Greubel, Oct 21 2019

CoefficientList[Series[(1+x-2*x^2)/(1-2*x)^2,{x,0,40}],x] (* Vincenzo Librandi, Jun 22 2012 *)
LinearRecurrence[{4,-4}, {1,5,14}, 40] (* G. C. Greubel, Oct 21 2019 *)

x='x+O('x^40); Vec((1+x-2*x^2)/(1-2*x)^2) \\ Altug Alkan, Mar 03 2018

[1]+[2^(n-1)*(2*n+3) for n in (1..40)] # G. C. Greubel, Oct 21 2019

G.f.: (1+x-2*x^2)/(1-2*x)^2.
a(n) = 4*(a(n-1) - a(n-2)).
a(n) = (n+1)*2^n + 2^(n-1), n > 0.
a(n) = A118413(n+1,n-1) for n > 2. - Reinhard Zumkeller, Apr 27 2006
E.g.f.: (1/2)*(-1 + exp(2*x)*(3 + 4*x)). - Stefano Spezia, Oct 22 2019
From Amiram Eldar, Oct 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 4*sqrt(2)*arcsinh(1) - 11/3.
Sum_{n>=0} (-1)^n/a(n) = 13/3 - 4*sqrt(2)*arccot(sqrt(2)). (End)

A118414 a(n) = (2n - 1) (2^n - 1).

Original entry on oeis.org

1, 9, 35, 105, 279, 693, 1651, 3825, 8687, 19437, 42987, 94185, 204775, 442341, 950243, 2031585, 4325343, 9175005, 19398619, 40894425, 85983191, 180355029, 377487315, 788529105, 1644167119, 3422552013, 7113539531, 14763950025, 30601641927, 63350767557, 130996502467, 270582939585

Offset: 1

Reinhard Zumkeller, Apr 27 2006

Row sums of triangle A118413.
For fixed n, define a triangle T(r,c) counting down the first n odd numbers on the left side, T(r,1) = 2*(n-r)+1, and counting up odd numbers on the right side, T(r,r) = 2*(n+r)-3, r>1. The interior elements are set by T(r,c)=T(r-1,c-1) + T(r-1,c). The sum of all members in this triangle is a(n). - J. M. Bergot, Oct 12 2012
Row sums of triangle A277046. - Miquel Cerda, Sep 28 2016

			The triangle T(r,c) for n=4 has row(1)=7; row(2) = 5, 9; row(3) = 3, 14, 11; row(4) = 1, 17, 25, 13, and a sum of 7+5+9+...+13 = 105 = a(4). - _J. M. Bergot_, Oct 12 2012

Altug Alkan, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4)

Cf. A014477, A000225, A005408, A118413, A277046.

[(2*n-1)*(2^n-1): n in [1..40]]; // Vincenzo Librandi, Dec 26 2010

Table[(2 n - 1) (2^n - 1), {n, 32}] (* or *)
Rest@ CoefficientList[Series[-x (-1 - 3 x + 6 x^2)/((2 x - 1)^2*(x - 1)^2), {x, 0, 32}], x] (* Michael De Vlieger, Sep 26 2016 *)
LinearRecurrence[{6,-13,12,-4},{1,9,35,105},40] (* Harvey P. Dale, Sep 12 2023 *)

a(n)=(2*n-1)*(2^n-1) \\ Charles R Greathouse IV, Oct 12 2012

a(n) = A005408(n-1)*(A000079(n) - 1). Corrected by Omar E. Pol, Sep 26 2016
G.f. -x*(-1-3*x+6*x^2) / ( (2*x-1)^2*(x-1)^2 ). - R. J. Mathar, Oct 15 2012
a(n) = A005408(n-1)*A000225(n). - Miquel Cerda, Sep 26 2016

A118415 a(n) = (4n - 3) 2^(n - 1).

Original entry on oeis.org

1, 10, 36, 104, 272, 672, 1600, 3712, 8448, 18944, 41984, 92160, 200704, 434176, 933888, 1998848, 4259840, 9043968, 19136512, 40370176, 84934656, 178257920, 373293056, 780140544, 1627389952, 3388997632, 7046430720, 14629732352, 30333206528, 62813896704

Offset: 1

Reinhard Zumkeller, Apr 27 2006

Central terms of the triangle in A118413.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A058962.

[(4*n-3)*2^(n-1): n in [1..40]]; // Vincenzo Librandi, Dec 26 2010

CoefficientList[Series[(1 + 6 x)/(-1 + 2 x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, May 21 2014 *)
LinearRecurrence[{4,-4},{1,10},30] (* Harvey P. Dale, Sep 16 2022 *)

a(n) = A016813(n-1)*A000079(n-1).

O.g.f.: x*(1+6*x)/(-1+2*x)^2 . - R. J. Mathar, Feb 26 2008

More terms from R. J. Mathar, Feb 26 2008

A219931 Coefficients related to an asymptotic expansion of the logarithm of the central binomial.

Original entry on oeis.org

1, 6, 5, 28, 9, 22, 13, 120, 17, 38, 21, 92, 25, 54, 29, 496, 33, 70, 37, 156, 41, 86, 45, 376, 49, 102, 53, 220, 57, 118, 61, 2016, 65, 134, 69, 284, 73, 150, 77, 632, 81, 166, 85, 348, 89, 182, 93, 1520, 97, 198, 101, 412, 105, 214, 109, 888, 113, 230, 117

Offset: 1

Peter Luschny, Dec 01 2012

An asymptotic expansion of the logarithm of the central binomial (A000984) for n>0 is given by log(binomial(2*n,n)) ~ (n*log(16)-log(Pi)-log(n) + sum_{k>=1}((-4)^(-k)*A002425(k)/a(k)*n^(1-2*k)))/2.

An asymptotic expansion of the logarithm of the swinging factorial (A056040) for n>1 is given by log(swing(n)) ~ (n*log(4)-log(Pi)-(-1)^n*(log(n/2) - (1/2)*sum_{k>=1}((-1)^k*A002425(k)/a(k)*n^(1-2*k))))/2.

			log(binomial(2*n,n)) = n*log(4) - (log(n)+log(Pi))/2 - 1/(8*a(1)*n) + 1/(32*a(2)*n^3) - 1/(128*a(3)*n^5) + 17/(512*a(4)*n^7) - 31/(2048*a(5)*n^9) + 691/(8192*a(6)*n^11) + O(1/n^13).
log(swing(n)) = n*log(2) - (1/2)*log(Pi) - (1/4)*(-1)^n*(2*log(n/2) + 1/(a(1)*n) - 1/(a(2)*n^3) + 1/(a(3)*n^5) - 17/(a(4)*n^7) + 31/(a(5)*n^9) - 691/(a(6)*n^11)) + O(1/n^13).

Peter Luschny, Table of n, a(n) for n = 1..300
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Cf. A006519, A118413, A385054.

Coeff_list := proc(len) local n;
asympt(ln(n/2)/2+lnGAMMA(n/2+1/2)-lnGAMMA(n/2+1),n,2*len+3);
subs(n=1/n,simplify(convert(%,polynom)));
[seq(4*coeff(unapply(%,n)(n),n,2*k+1),k=0..len-1)] end:
A219931_list := n -> denom(Coeff_list(n)); A219931_list(59);

max = 60; s = Normal[Series[Log[x/2]/2+LogGamma[x/2+1/2]-LogGamma[x/2+1], {x, Infinity, 2*max}]] /. x -> 1/x; a[n_] := Denominator[4*Coefficient[s, x^(2*n-1), 1]]; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Feb 17 2014 *)
a[n_] := Denominator[2*EulerE[2*n-1, 1]/(2*n-1)]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Apr 04 2014, after Peter Luschny *)

a(n) = denominator(2*E(2*n-1, 1)/(2*n-1)) where E(n, x) is the Euler polynomial. - Peter Luschny, Apr 03 2014

Warning: a(n) != (2*n-1)*2^valuation(n, 2). This was mistakenly assumed several times in the past, see A385054. - Peter Luschny, Jun 17 2025

Edited and incorrect entries removed by Georg Fischer and Peter Luschny, Jun 16 2025

A277046 Triangle read by rows: T(n,k) = 2^n - n + k - 1 for n >= 1, with 1 <= k <= 2n-1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 27, 28, 29, 30, 31, 32, 33, 34, 35, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519

Offset: 1

Miquel Cerda, Sep 27 2016

			Triangle begins:
1;
2,   3,  4;
5,   6,  7,  8,  9;
12, 13, 14, 15, 16, 17, 18;
27, 28, 29, 30, 31, 32, 33, 34, 35;
58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68;
...
Written as an isosceles triangle the sequence begins:
.                      1;
.                  2,  3,  4;
.              5,  6,  7,  8,  9;
.         12, 13, 14, 15, 16, 17, 18;
.     27, 28, 29, 30, 31, 32, 33, 34, 35;
. 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68;
..

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Row lengths are A005408.
Row sums give A118414.
Column 1 gives A000325, n>=1.
Middle diagonal gives A000225.
Right border gives A083706.
Cf. A118413.

Table[2^n-n+k-1,{n,10},{k,2n-1}]//Flatten (* Harvey P. Dale, Nov 27 2021 *)

Definition from Omar E. Pol, Sep 28 2016

cp's OEIS Frontend

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

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A117303 Self-inverse permutation of the natural numbers based on the bijection (2*x-1)*2^(y-1) <--> (2*y-1)*2^(x-1).

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A052951 Expansion of (1 + x - 2*x^2)/(1 - 2*x)^2.

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A118414 a(n) = (2*n - 1) * (2^n - 1).

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A118415 a(n) = (4*n - 3) * 2^(n - 1).

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A219931 Coefficients related to an asymptotic expansion of the logarithm of the central binomial.

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

A117303 Self-inverse permutation of the natural numbers based on the bijection (2x-1)2^(y-1) <--> (2y-1)2^(x-1).

A052951 Expansion of (1 + x - 2x^2)/(1 - 2x)^2.

A118414 a(n) = (2n - 1) (2^n - 1).

A118415 a(n) = (4n - 3) 2^(n - 1).