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a(n) = T(n,n+2), array T as in A050143.
9

%I #54 Oct 23 2023 12:37:18

%S 0,1,6,33,180,985,5418,29953,166344,927441,5188590,29113953,163786428,

%T 923511849,5217709266,29532022785,167417253648,950453221153,

%U 5402869685334,30748881013153,175186193208900,999071379620601,5702750629608186,32578618535692033,186257611786501080

%N a(n) = T(n,n+2), array T as in A050143.

%C Form an array having the first column all 1's and the first row the squares: m(n,1) = 1 and m(1,n) = n^2 for n = 1, 2, 3, .... Define interior terms m(i,j) = m(i,j-1) + m(i-1,j-1) + m(i-1,j). Then the terms on the main diagonal are the terms of this sequence. - _J. M. Bergot_, Nov 16 2012

%C Form an array with first row m(1,j)=1 and first column m(n,1) = n*(n-1)+1 for n=1,2,3... The remaining terms m(i,j) = m(i,j-1) + m(i-1,j-1) + m(i-1,j); Sum_{n=1,2,3,...} T(n,n) = a(n). The first five terms in the main diagonal are 1, 5, 27, 147, 805 with partial sums 1, 6, 33, 180, 985. - _J. M. Bergot_, Jan 26 2013

%H Vincenzo Librandi, <a href="/A050151/b050151.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%F From _Vladeta Jovovic_, Mar 28 2004: (Start)

%F G.f.: ((1-3*x)/sqrt(1-6*x+x^2)-1)/(4*x).

%F E.g.f.: exp(3*x)*BesselI(1, 2*sqrt(2)*x)/sqrt(2). (End)

%F a(n) = Sum_{k=0..n} binomial(n, k)*binomial(n+1, k+1)/2. - _Paul Barry_, Sep 20 2004

%F a(n) = n*R(n)/2, where R(n)=A006318(n) are the large Schroeder numbers. - _Emeric Deutsch_, Jul 14 2005

%F From _David Callan_, Aug 16 2006: (Start)

%F a(n) = Sum_{k=0..n} 2^(k-1)*binomial(n, k)*binomial(n, k-1).

%F a(n) = (CentralDelannoy(n+1) - 3*CentralDelannoy(n))/4 where CentralDelannoy(n) is A001850. (End)

%F a(n) = (1/Pi)*Integral_{x=3-2*sqrt(2)..3+2*sqrt(2)} x^n*(x-3)/(4*sqrt(-x^2+6x-1)). - _Paul Barry_, Sep 16 2006

%F D-finite with recurrence (n+1)*a(n) - 9*n*a(n-1) + (19*n-27)*a(n-2) + 3*(-n+2)*a(n-3) = 0. - _R. J. Mathar_, Nov 16 2012

%F a(n) ~ sqrt(3*sqrt(2)-4) * (1+sqrt(2))^(2*n+2) / (4*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 13 2016

%F a(n) = a(n)*(+9*a(n+1) - 90*a(n+2) + 81*a(n+3) - 12*a(n+4)) + a(n+1)*(-24*a(n+1) + 334*a(n+2) - 408*a(n+3) + 65*a(n+4)) + a(n+2)*(+72*a(n+2) + 54*a(n+3) - 18*a(n+4)) + a(n+3)*(+a(n+4)) for all n in Z where a(n) = -(1/2) * A050146(-n) if n < 0. - _Michael Somos_, Nov 25 2016

%F From _Peter Luschny_, Nov 09 2017: (Start)

%F a(n) = (-1)^n*n*hypergeom([-n, n+1], [2], 2).

%F a(n) = n*A001003(n). (End)

%e G.f. = x + 6*x^2 + 33*x^3 + 180*x^4 + 985*x^5 + 5418*x^6 + 29953*x^7 + ...

%p A050151 := n -> (-1)^n*n*hypergeom([-n, n+1], [2], 2):

%p seq(simplify(A050151(n)), n=0..21); # _Peter Luschny_, Nov 09 2017

%t CoefficientList[Series[((1 - 3*x)/(1 - 6*x + x^2)^(1/2) - 1)/(4*x), {x, 0, 100}], x] (* _Vincenzo Librandi_, Feb 02 2013 *)

%t a[ n_] := n / 2 Hypergeometric2F1[1 + n, -n, 2, -1]; (* _Michael Somos_, Nov 25 2016 *)

%o (PARI) a(n) = n*hypergeom([-n, n+1], [2], 2)\/(-1)^n \\ _Charles R Greathouse IV_, Oct 23 2023

%Y Cf. A001003, A001850, A006318, A050143, A050146.

%K nonn

%O 0,3

%A _Clark Kimberling_