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A050146 a(n) = T(n,n), array T as in A050143. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

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

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.

Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.

Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374.

FORMULA

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. - Vladeta Jovovic, Mar 31 2004

(n-1)*(n-2)*a(n) = 3*(2*n-3)*(n-2)*a(n-1)-(n-1)*(n-3)*a(n-2), 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) = n*hypergeometric([1-n, n], [2], -1) for n>=1. - Peter Luschny, Sep 17 2014

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

EXAMPLE

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

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, Hypergeometric2F1[1 - n, n, 2, -1]; (* Michael Somos, Nov 25 2016 *)

PROG

(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

(Haskell)

a050146 n = if n == 0 then 1 else a035607 (2 * n - 2) (n - 1)

-- Reinhard Zumkeller, Nov 05 2013, Jul 20 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

(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

CROSSREFS

Cf. A002003, A050151.

Cf. A006318. - Abdullahi Umar, Aug 25 2008

Sequence in context: A298646 A199309 A083325 * A083879 A081671 A244785

Adjacent sequences:  A050143 A050144 A050145 * A050147 A050148 A050149

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified November 12 09:29 EST 2019. Contains 329054 sequences. (Running on oeis4.)