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A026000 a(n) = T(2n, n), where T is the Delannoy triangle (A008288). 3
1, 5, 41, 377, 3649, 36365, 369305, 3800305, 39490049, 413442773, 4354393801, 46082942185, 489658242241, 5220495115997, 55818956905529, 598318746037217, 6427269150511105, 69175175263888037, 745778857519239785 (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

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

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)

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.

Sequence in context: A145215 A083884 A156153 * A058475 A199684 A177506

Adjacent sequences:  A025997 A025998 A025999 * A026001 A026002 A026003

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified September 18 20:51 EDT 2019. Contains 327181 sequences. (Running on oeis4.)