A050151 a(n) = T(n,n+2), array T as in A050143.

0, 1, 6, 33, 180, 985, 5418, 29953, 166344, 927441, 5188590, 29113953, 163786428, 923511849, 5217709266, 29532022785, 167417253648, 950453221153, 5402869685334, 30748881013153, 175186193208900, 999071379620601, 5702750629608186, 32578618535692033, 186257611786501080

Offset

0,3

Comments

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

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Milan Janjic, Two Enumerative Functions

Formula

From Vladeta Jovovic, Mar 28 2004: (Start)

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

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

a(n) = Sum_{k=0..n} binomial(n, k)*binomial(n+1, k+1)/2. - Paul Barry, Sep 20 2004

a(n) = n*R(n)/2, where R(n)=A006318(n) are the large Schroeder numbers. - Emeric Deutsch, Jul 14 2005

From David Callan, Aug 16 2006: (Start)

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

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

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

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

a(n) ~ sqrt(3*sqrt(2)-4) * (1+sqrt(2))^(2*n+2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 13 2016

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

From Peter Luschny, Nov 09 2017: (Start)

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

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

Example

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

Maple

A050151 := n -> (-1)^n*n*hypergeom([-n, n+1], [2], 2):
seq(simplify(A050151(n)), n=0..21); # Peter Luschny, Nov 09 2017

Mathematica

CoefficientList[Series[((1 - 3*x)/(1 - 6*x + x^2)^(1/2) - 1)/(4*x), {x, 0, 100}], x] (* Vincenzo Librandi, Feb 02 2013 *)
a[ n_] := n / 2 Hypergeometric2F1[1 + n, -n, 2, -1]; (* Michael Somos, Nov 25 2016 *)

Prog

(PARI) a(n) = n*hypergeom([-n, n+1], [2], 2)\/(-1)^n \\ Charles R Greathouse IV, Oct 23 2023

Crossrefs

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

Sequence in context: A092851 A137627 A138395 * A009162 A012718 A297221

Adjacent sequences: A050148 A050149 A050150 * A050152 A050153 A050154

Keyword

nonn

Author

Clark Kimberling

Status

approved