A303602 -id:A303602 - cp's OEIS frontend

A260006 a(n) = f(1,n,n), where f is the Sudan function defined in A260002.

0, 3, 12, 35, 90, 217, 504, 1143, 2550, 5621, 12276, 26611, 57330, 122865, 262128, 557039, 1179630, 2490349, 5242860, 11010027, 23068650, 48234473, 100663272, 209715175, 436207590, 905969637, 1879048164, 3892314083, 8053063650, 16642998241, 34359738336

Offset: 0

Natan Arie Consigli, Jul 23 2015

f(1,n,n) = 2^n*(n+2) - (n+2) = (2^n - 1)*(n+2).
To evaluate the Sudan function see A260002 and A260003.
The numbers are alternately even and odd because for even n (2^n-1)*(n+2) is even and (2^(n+1)-1)*(n+1+2) is odd.
From Enrique Navarrete, Oct 02 2021: (Start)
a(n-2) is the number of ways we can write [n] as the union of 2 sets of sizes i, j which intersect in exactly 1 element (1 < i, j < n;  i = j allowed), n>=2.
For n = 4, a(n-2) = 12 since [4] can be written as the unions:
{1,2} U {1,3,4}; {2,3} U {1,2,4}; {1,2} U {2,3,4}; {2,3} U {1,3,4};
{1,3} U {1,2,4}; {2,4} U {1,2,3}; {1,3} U {2,3,4}; {2,4} U {1,3,4};
{1,4} U {1,2,3}; {3,4} U {1,2,3}; {1,4} U {2,3,4}; {3,4} U {1,2,4}. (End)

			a(4) = (2^4 - 1)*(4 + 2) = 90.

Colin Barker, Table of n, a(n) for n = 0..1000
Wikipedia, Sudan function (see diagonal of "Values of F1(x, y)" table).
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A000295 (f(1,0,n)), A000325 (f(1,2,n)), A005408 (f(1,n,1) = 2n+1), A001787 (n*2^(n-1)), A079583 (f(1,1,n)), A123720 (f(1,4,n)), A133124 (f(1,3,n)).
Cf. A260002, A260003, A260004, A260005.
Cf. A000984, A002697, A303602.

[(2^n-1)*(n+2): n in [0..30]]; // Vincenzo Librandi, Aug 22 2015

Table[(2^n -1)(n+2), {n, 0, 30}] (* Michael De Vlieger, Aug 22 2015 *)
CoefficientList[Series[x(3 -6x +2x^2)/((1-x)^2 (1-2x)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 22 2015 *)
LinearRecurrence[{6,-13,12,-4},{0,3,12,35},40] (* Harvey P. Dale, Mar 04 2023 *)

vector(40, n, n--; (2^n-1)*(n+2)) \\ Michel Marcus, Jul 29 2015

concat(0, Vec(x*(3-6*x+2*x^2)/((1-x)^2*(1-2*x)^2) + O(x^40))) \\ Colin Barker, Jul 29 2015

[(n+2)*(2^n -1) for n in (0..30)] # G. C. Greubel, Dec 30 2021

a(n) = (2^n -1)*(n+2).
a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4) for n>3. - Colin Barker, Jul 29 2015
G.f.: x*(3 - 6*x + 2*x^2) / ((1-x)^2*(1-2*x)^2). - Colin Barker, Jul 29 2015
E.g.f.: 2*(x+1)*exp(2*x) - (x+2)*exp(x). - Robert Israel, Aug 23 2015
From Enrique Navarrete, Oct 02 2021: (Start)
a(n-2) = Sum_{j=2..n/2} binomial(n,j)*j, n even > 2.
a(n-2) = (Sum_{j=2..floor(n/2)} binomial(n,j)*j) + (1/2)*binomial(n, ceiling(n/2))*ceiling(n/2), n odd > 1. (End)
From Wolfdieter Lang, Nov 12 2021: (Start)
The previous bisection becomes for a(n):
a(2*k) = 2*(A002697(k+1) - (k+1)), and a(2*k+1) = A303602(k+1) - (2*k+3)*(2 - A000984(k+1))/2 = (2*k+3)*(4^(k+1) - 2)/2, for k >= 0. (End)

A130783 Maximum value of the n-th difference of a permutation of 0..n.

Original entry on oeis.org

0, 1, 3, 10, 25, 66, 154, 372, 837, 1930, 4246, 9516, 20618, 45332, 97140, 210664, 447661, 960858, 2028478, 4319100, 9070110, 19188796, 40122028, 84438360, 175913250, 368603716, 765561564, 1598231992, 3310623412, 6889682280, 14238676712, 29551095248

Offset: 0

R. H. Hardin, Aug 19 2007

For n>1, a(n) is also the maximum value of the n-th difference of a permutation of 1..n. - Michel Marcus, Apr 15 2017

			a(1)=1 because 0 1 has a first difference of 1;
a(2)=3 because 2 0 1 has a second difference of 3.

Fung Lam, Table of n, a(n) for n = 0..3000
F. Disanto, A. Frosini, and S. Rinaldi, Square involutions, J. Int. Seq. 14 (2011) # 11.3.5.

Cf. A000346, A033504.

A130783:=n->(n+1)*(2^n-binomial(n,floor(n/2)))/2; seq(A130783(n), n=0..50); # Wesley Ivan Hurt, Nov 25 2013

Table[(n + 1) (2^n - Binomial[n, Floor[n/2]])/2, {n, 0, 50}] (* Wesley Ivan Hurt, Nov 25 2013 *)

a(n)=(n+1)*(2^n-binomial(n,n\2))/2 \\ Charles R Greathouse IV, Jan 30 2012

from math import comb
def A130783(n): return (n+1)*((1<>1))>>1 # Chai Wah Wu, Jun 04 2024

a(n) = (n+1)*(2^(n-1)-binomial(n-1,n/2)) if n is even else ((n+1)/2)*(2^n-binomial(n,(n+1)/2)). - Vladeta Jovovic, Aug 23 2007
a(n) = (n+1)*(2^n-binomial(n,[n/2]))/2, where [x] is floor. - Graeme McRae, Jan 30 2012
G.f.: (1-sqrt((1-2*x)/(1+2*x)))/(2*(1-2*x)^2). - Vladeta Jovovic, Aug 24 2007
Asymptotics: a(n) ~ 2^(n-1)*(n+1-sqrt(2*n/Pi)). - Fung Lam, Mar 28 2014
D-finite with recurrence (n-1)*n*a(n) = 2*(n-1)*(n+1)*a(n-1) + 4*(n-2)*n*a(n-2) - 8*(n-1)*n*a(n-3). - Vaclav Kotesovec, Mar 28 2014
a(2n) = A303602(n). a(2n+1) = A033504(n). - R. J. Mathar, Nov 20 2020

cp's OEIS Frontend

A260006 a(n) = f(1,n,n), where f is the Sudan function defined in A260002.

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A130783 Maximum value of the n-th difference of a permutation of 0..n.

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