A200375 -id:A200375

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A200312

a(n) = A000108(n)*A006130(n), where A000108 is the Catalan numbers and A006130(n) = A006130(n-1) + 3*A006130(n-2).

+10
2

1, 1, 8, 35, 266, 1680, 12804, 93093, 726440, 5635058, 45063668, 362121760, 2955642508, 24284658100, 201428123040, 1680921310635, 14119413718770, 119205791509200, 1011387051005100, 8617021562542470, 73704123363739440, 632601537174078420

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

OFFSET

0,3

COMMENTS

More generally, given {S} such that: S(n) = b*S(n-1) + c*S(n-2), S(0)=1, |b|>0, |c|>0, then Sum_{n>=0} S(n)*Catalan(n)*x^n = sqrt( (1-2*b*x - sqrt(1-4*b*x-16*c*x^2))/(2*b^2+8*c) )/x.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..450

S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)

FORMULA

G.f.: sqrt( (1-2*x - sqrt(1-4*x-48*x^2))/26 )/x.

G.f.: (1/x)*Series_Reversion( x*sqrt(1-12*x^2) - x^2 ).

G.f.: (1/x)*Series_Reversion( x-x^2 - 6*x^3*Sum_{n>=0} A000108(n)*3^n*x^(2*n) ).

G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x)^2 + 13*x^2*A(x)^4).

Conjecture: n*(n+1)*a(n) -2*n*(2*n-1)*a(n-1) -12*(2*n-1)*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 17 2011

a(n) = ( ((1+sqrt(13))/2)^(n+1) - ((1-sqrt(13))/2)^(n+1) )/sqrt(13) * binomial(2*n+1,n)/(2*n+1). - Paul D. Hanna, Sep 25 2012

0 = +a(n)*(+110592*a(n+3) -9216*a(n+4) -7392*a(n+5) +858*a(n+6)) +a(n+1)*(+6912*a(n+3) -1968*a(n+4) -910*a(n+5) +154*a(n+6)) +a(n+2)*(-240*a(n+3) -2*a(n+4) +41*a(n+5) -4*a(n+6)) +a(n+3)*(+6*a(n+3) +5*a(n+4) +3*a(n+5) -a(n+6)) for all n in Z. - Michael Somos, Jul 28 2018

EXAMPLE

G.f.: A(x) = 1 + x + 2*4*x^2 + 5*7*x^3 + 14*19*x^4 + 42*40*x^5 + 132*97*x^6 + 429*217*x^7 + ... + A000108(n)*A006130(n)*x^n + ...

where the g.f. of A006130, F(x) = 1/(1-x-3*x^2), begins:

F(x) = 1 + x + 4*x^2 + 7*x^3 + 19*x^4 + 40*x^5 + 97*x^6 + 217*x^7 + ...

MATHEMATICA

CoefficientList[Series[Sqrt[(1 - 2*x - Sqrt[1 - 4*x - 48*x^2])/26]/x, {x, 0, 30}], x] (* G. C. Greubel, Jul 27 2018 *)

PROG

(PARI) {a(n)=binomial(2*n, n)/(n+1)*polcoeff(1/(1-x-3*x^2+x*O(x^n)), n)}

(PARI) {a(n)=polcoeff(sqrt((1-2*x - sqrt(1-4*x-48*x^2+x^3*O(x^n)))/26)/x, n)}

(PARI) {a(n)=polcoeff(serreverse(x*sqrt(1-12*x^2+x^2*O(x^n)) - x^2)/x, n)}

(PARI) {a(n)=polcoeff((1/x)*serreverse(x-x^2 - 6*x^3*sum(m=0, n\2, binomial(2*m, m)/(m+1)*3^m*x^(2*m))+x^3*O(x^n)), n)}

(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt((1-2*x - Sqrt(1-4*x-48*x^2))/26)/x)); // G. C. Greubel, Jul 27 2018

CROSSREFS

Cf. A098614, A098616, A200375.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 16 2011

STATUS

approved

A200376

G.f.: 1/sqrt(1-10*x^2 + x^4/(1-8*x^2)) + x/(1-9*x^2).

+10
2

1, 1, 5, 9, 37, 81, 301, 729, 2549, 6561, 22045, 59049, 193029, 531441, 1703469, 4782969, 15111573, 43046721, 134539837, 387420489, 1200901157, 3486784401, 10739313997, 31381059609, 96172251061, 282429536481, 862142190941, 2541865828329, 7734936371269, 22876792454961, 69439155241581

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

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..30.

FORMULA

D-finite with recurrence: n*a(n) +(n-1)*a(n-1) +(24-17*n)*a(n-2) +(41-17*n)*a(n-3) +72*(n-3)*a(n-4) +72*(n-4)*a(n-5)=0. - R. J. Mathar, Nov 17 2011

G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x) + 9*x^2*A(x)^2). - Paul D. Hanna, Nov 18 2014

Let G(x) = g.f. of A200375, then g.f. A(x) satisfies:

(1) A(x) = x/Series_Reversion(x*G(x)),

(2) A(x) = G(x/A(x)) and G(x) = A(x*G(x)),

where A200375(n) = A000108(n)*A001045(n), the product of Catalan and Jacobsthal numbers.

a(n) ~ 3^(n-1). - Vaclav Kotesovec, Jun 29 2013

EXAMPLE

G.f.: A(x) = 1 + x + 5*x^2 + 9*x^3 + 37*x^4 + 81*x^5 + 301*x^6 + 729*x^7 +...

The g.f. of A200375(n) = A000108(n)*A001045(n) begins:

G(x) = 1 + x + 2*3*x^2 + 5*5*x^3 + 14*11*x^4 + 42*21*x^5 + 132*43*x^6 +...

where A(x) = G(x/A(x)) and G(x) = A(x*G(x)).

MATHEMATICA

CoefficientList[Series[1/Sqrt[1-10x^2+x^4/(1-8x^2)]+x/(1-9x^2), {x, 0, 30}], x] (* Harvey P. Dale, Nov 19 2011 *)

PROG

(PARI) {a(n)=polcoeff(1/sqrt(1-10*x^2 + x^4/(1-8*x^2 +x*O(x^n))) + x/(1-9*x^2 +x*O(x^n)), n)}

for(n=0, 30, print1(a(n), ", "))

(PARI) {a(n)=local(G=sum(m=0, n, binomial(2*m, m)/(m+1)*polcoeff(1/(1-x-2*x^2+x*O(x^m)), m)*x^m)+x*O(x^n)); polcoeff(x/serreverse(x*G), n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A200375, A098615, A098617.

KEYWORD

nonn,changed

AUTHOR

Paul D. Hanna, Nov 16 2011

STATUS

approved

A200538

Product of Jacobsthal and Motzkin numbers: a(n) = A001045(n+1)*A001006(n).

+10
2

1, 1, 6, 20, 99, 441, 2193, 10795, 55233, 284735, 1494404, 7914270, 42360541, 228460935, 1241224182, 6784445340, 37288826697, 205937705799, 1142317727466, 6361104740100, 35548154733969, 199295884785459, 1120615326442269, 6318077793648075, 35710056983891367, 202297486497822121

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

OFFSET

0,3

COMMENTS

The g.f. for the Jacobsthal numbers is 1/(1-x-2*x^2) and the g.f. M(x) for the Motzkin numbers satisfy: M(x) = 1 + x*M(x) + x^2*M(x)^2.

LINKS

Table of n, a(n) for n=0..25.

EXAMPLE

G.f.: A(x) = 1 + x + 6*x^2 + 20*x^3 + 99*x^4 + 441*x^5 + 2193*x^6 +...

where A(x) = 1*1 + 1*1*x + 3*2*x^2 + 5*4*x^3 + 11*9*x^4 + 21*21*x^5 + 43*51*x^6 + 85*127*x^7 + 171*323*x^8 +...+ A001045(n+1)*A001006(n)*x^n +...

PROG

(PARI) {A001006(n)=polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2), n)}

{A001045(n)=polcoeff( x/(1-x-2*x^2+x*O(x^n)), n)}

{a(n)=A001045(n+1)*A001006(n)}

CROSSREFS

Cf. A200375, A200539, A200540, A001045, A001006.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 18 2011

STATUS

approved

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