A050146 a(n) = T(n,n), array T as in A050143.

1, 1, 4, 18, 88, 450, 2364, 12642, 68464, 374274, 2060980, 11414898, 63521352, 354870594, 1989102444, 11180805570, 63001648608, 355761664002, 2012724468324, 11406058224594, 64734486343480, 367891005738690, 2093292414443164, 11923933134635298, 67990160422313808

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

Also main diagonal of array : m(i,1)=1, i>=1; m(1,j)=2, j>1; m(i,j)=m(i,j-1)+m(i-1,j-1)+m(i-1,j): 1 2 2 2 ... / 1 4 8 12 ... / 1 6 18 38 ... / 1 8 32 88 ... / - Benoit Cloitre, Aug 05 2002

a(n) is also the number of order-preserving partial transformations (of an n-element chain) of waist n (waist(alpha) = max(Im(alpha))). - Abdullahi Umar, Aug 25 2008

Define a finite triangle T(r,c) with T(r,0) = binomial(n,r) for 0<=r<=n, and the other terms recursively with T(r,c) = T(r,c-1) + 2*T(r-1,c-1). The sum of the last terms in each row is Sum_{r=0..n} T(r,r)=a(n+1). For n=4 the triangle is 1; 4 6; 6 14 26; 4 16 44 96; 1 9 41 129 321 with the sum of the last terms being 1 + 6 + 26 + 96 + 321 = 450 = a(5). - J. M. Bergot, Jan 29 2013

It may be better to define a(0) = 0 for formulas without exceptions. - Michael Somos, Nov 25 2016

a(n) is the number of points at L1 distance n-1 from any point in Z^n, for n>=1. - Shel Kaphan, Mar 24 2023

Examples

			G.f. = 1 + x + 4*x^2 + 18*x^3 + 88*x^4 + 450*x^5 + 2364*x^6 + 12642*x^7 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra, 278 (2004), 342-359.
• A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq., 7 (2004), 04.3.8.
• Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 4.
• Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374.

Crossrefs

Cf. A002003, A006318, A050151, A008288, A002002, A026002.

-1-diagonal of A266213 for n>=1.

Programs

Haskell

a050146 n = if n == 0 then 1 else a035607 (2 * n - 2) (n - 1)
-- Reinhard Zumkeller, Nov 05 2013, Jul 20 2013

Mathematica

Flatten[{1,RecurrenceTable[{(n-3)*(n-1)*a[n-2]-3*(n-2)*(2*n-3)*a[n-1]+(n-2)*(n-1)*a[n]==0,a[1]==1,a[2]==4},a,{n,20}]}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[ n_] := If[ n == 0, 1, Sum[ Binomial[n, k] Binomial[n + k - 2, k - 1], {k, n}]]; (* Michael Somos, Nov 25 2016 *)
a[ n_] := If[ n == 0, 1, n Hypergeometric2F1[1 - n, n, 2, -1]]; (* Michael Somos, Nov 25 2016 *)

Maxima

taylor(-(x^4+sqrt(x^2-6*x+1)*(x^3-5*x^2+5*x+1)-8*x^3+16*x^2-6*x+1)/(x^3+sqrt(x^2-6*x+1)*(x^2-4*x-1)-7*x^2+7*x-1),x,0,10); /* Vladimir Kruchinin, Nov 25 2016 */

PARI

a(n)=if(n==0, 1, sum(k=1,n, binomial(n, k)*binomial(n+k-2, k-1)) ); \\ Joerg Arndt, May 04 2013

Sage

A050146 = lambda n : n*hypergeometric([1-n, n], [2], -1) if n>0 else 1
[round(A050146(n).n(100)) for n in (0..24)] # Peter Luschny, Sep 17 2014

Formula

From Vladeta Jovovic, Mar 31 2004: (Start)

Coefficient of x^(n-1) in expansion of ((1+x)/(1-x))^n, n > 0.

a(n) = Sum_{k=1..n} binomial(n, k)*binomial(n+k-2, k-1), n > 0. (End)

D-finite with recurrence (n-1)*(n-2)*a(n) = 3*(2*n-3)*(n-2)*a(n-1) - (n-1)*(n-3)*a(n-2) for n > 2. - Vladeta Jovovic, Jul 16 2004

a(n+1) = Jacobi_P(n, 1, -1, 3); a(n+1) = Sum{k=0..n} C(n+1, k)*C(n-1, n-k)*2^k. - Paul Barry, Jan 23 2006

a(n) = n*A006318(n-1) - Abdullahi Umar, Aug 25 2008

a(n) ~ sqrt(3*sqrt(2)-4)*(3+2*sqrt(2))^n/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012

a(n+1) = A035607(2*n,n). - Reinhard Zumkeller, Jul 20 2013

a(n) = n*hypergeometric([1-n, n], [2], -1) for n >= 1. - Peter Luschny, Sep 17 2014

O.g.f.: -(x^4 + sqrt(x^2 - 6*x + 1)*(x^3 - 5*x^2 + 5*x + 1) - 8*x^3 + 16*x^2 - 6*x + 1)/(x^3 + sqrt(x^2 - 6*x + 1)*(x^2 - 4*x - 1)- 7*x^2 + 7*x - 1). - Vladimir Kruchinin, Nov 25 2016

0 = a(n)*(a(n+1) - 18*a(n+2) + 65*a(n+3) - 12*a(n+4)) + a(n+1)*(54*a(n+2) - 408*a(n+3) + 81*a(n+4)) + a(n+2)*(72*a(n+2) + 334*a(n+3) - 90*a(n+4)) + a(n+3)*(-24*a(n+3) + 9*a(n+4)) for all integer n if a(0) = 0 and a(n) = -2*A050151(-n) for n < 0. - Michael Somos, Nov 25 2016

O.g.f: (2 - x + x*(3 - x)/sqrt(x^2 - 6*x + 1))/2. - Petros Hadjicostas, Feb 14 2021

a(n) = A002002(n) - A026002(n-1) for n>=2. - Shel Kaphan, Mar 24 2023