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

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)
\\ test: y=Ser(seq(303), 'x); 0 == 4*(x^2 + 11*x - 1)*y^3 + (x + 3)*y + 1
\\ Gheorghe Coserea, Aug 31 2018

Crossrefs

Cf. A008288, A027307, A026001.

Sequence in context: A145215 A083884 A156153 * A058475 A199684 A177506

Adjacent sequences: A025997 A025998 A025999 * A026001 A026002 A026003

Keyword

nonn,easy,changed

Author

Clark Kimberling

Status

approved