BBC Russian

A026000

a(n) = T(2n, n), where T is the Delannoy triangle (A008288).

1, 5, 41, 377, 3649, 36365, 369305, 3800305, 39490049, 413442773, 4354393801, 46082942185, 489658242241, 5220495115997, 55818956905529, 598318746037217, 6427269150511105, 69175175263888037, 745778857519239785, 8052432236270744665, 87063177396677721409

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Even order terms in the diagonal of rational function 1/(1 - (x + y^2 + x*y^2)). - Gheorghe Coserea, Aug 31 2018

LINKS

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

Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).

FORMULA

a(n) = ((2*n+3)*(n+1)*A027307(n+1)/2-(3*n+2)*n*A027307(n)) / (5*n+3) (guessed). - Mark van Hoeij, Jul 02 2010

Recurrence: 2*n*(2*n-1)*a(n) = (46*n^2-51*n+15)*a(n-1) - (18*n^2-82*n+85)*a(n-2) - (n-2)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 08 2012

a(n) ~ sqrt(150+70*sqrt(5))*((11+5*sqrt(5))/2)^n/(20*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012. Equivalently, a(n) ~ phi^(5*n + 2) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021

a(n) = hypergeom([-n, -n, n + 1], [1/2, 1], 1). - Peter Luschny, Mar 14 2018

From Gheorghe Coserea, Aug 31 2018:(Start)

G.f.: 1 + serreverse((-(44*x^2 + 88*x + 45) + (10*x + 9)*sqrt(20*x^2 + 44*x + 25))/(8*(x + 1)^2)).

G.f. y=A(x) satisfies:

0 = 4*(x^2 + 11*x - 1)*y^3 + (x + 3)*y + 1.

0 = 2*x*(x - 2)*(x^2 + 11*x - 1)*y'' + (5*x^3 + 8*x^2 - 87*x + 2)*y' + (x^2 - 7*x - 10)*y. (End)

From Peter Bala, Jan 20 2020: (Start)

a(n) = Sum_{k = 0..n} C(2*n, n-k) * C(2*n+k, k).

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

n*(2*n-1)*(10*n-13)*a(n) = (220*n^3-506*n^2+334*n-63*n)*a(n-1) + (n-1)*(2*n-3)*(10*n-3)*a(n-2). (End)

From Peter Bala, Apr 15 2023: (Start)

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

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

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

EXAMPLE

A(x) = 1 + 5*x + 41*x^2 + 377*x^3 + 3649*x^4 + 36365*x^5 + ...

MATHEMATICA

Flatten[{1, RecurrenceTable[{2*n*(2*n-1)*a[n] == (46*n^2-51*n+15)*a[n-1] - (18*n^2-82*n+85)*a[n-2] - (n-2)*(2*n-5)*a[n-3], a[1]==5, a[2]==41, a[3]==377}, a, {n, 20}]}] (* Vaclav Kotesovec, Oct 08 2012 *)

a[n_] := HypergeometricPFQ[{-n, -n, n + 1}, {1/2, 1}, 1];

Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 14 2018 *)

PROG

(PARI)

seq(N) = {

my(a = vector(N)); a[1]=5; a[2]=41; a[3]=377;

for (n=4, N,

a[n] = (46*n^2-51*n+15)*a[n-1] - (18*n^2-82*n+85)*a[n-2] - (n-2)*(2*n-5)*a[n-3];

a[n] /= 2*n*(2*n-1));

concat(1, a);

};

seq(18)