A098617 -id:A098617

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A098616

Product of Pell and Catalan numbers: a(n) = A000129(n+1)*A000108(n).

+10
12

1, 2, 10, 60, 406, 2940, 22308, 175032, 1408550, 11561836, 96425836, 814773960, 6960289532, 60012947800, 521582661000, 4564643261040, 40190674554630, 355772529165900, 3164408450118300, 28266363849505320, 253466716153665300, 2280803103062033160, 20588945107316958840

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

OFFSET

0,2

COMMENTS

Radius of convergence: r = (sqrt(2)-1)/4, where A(r) = sqrt(2+sqrt(2)).

More generally, given {S} such that: S(n) = b*S(n-1) + c*S(n-2), |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

Harvey P. Dale, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: A(x) = sqrt( (1-4*x - sqrt(1-8*x-16*x^2))/16 )/x.

Run lengths of zeros (mod 10) equal (5^k - (-1)^k)/2 - 1 starting at index (5^k + (-1)^k)/2:

a(n) == 0 (mod 10) for n = (5^k + (-1)^k)/2 through n = 5^k - 1 when k>=1.

a(n) ~ 2^(2*n-3/2) * (1+sqrt(2))^(n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, May 09 2014

A(-x) = 1/x * series reversion( x*(2*x + sqrt(1 - 4*x^2)) ). Compare with the o.g.f. B(x) of the central binomial numbers A000984, which satisfies B(-x) = 1/x * series reversion( x*(2*x + sqrt(1 + 4*x^2)) ). See also A214377. - Peter Bala, Oct 19 2015

n*(n+1)*a(n) -4*n*(2*n-1)*a(n-1) -4*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 17 2018

Sum_{n>=0} a(n)/16^n = 2*sqrt(3-sqrt(7)). - Amiram Eldar, May 05 2023

EXAMPLE

Sequence begins: [1*1, 2*1, 5*2, 12*5, 29*14, 70*42, 169*132, 408*429,...].

MATHEMATICA

With[{nn=30}, Times@@@Thread[{LinearRecurrence[{2, 1}, {1, 2}, nn], CatalanNumber[ Range[0, nn-1]]}]] (* Harvey P. Dale, Jan 04 2012 *)

a[n_] := Fibonacci[n + 1, 2] * CatalanNumber[n]; Array[a, 25, 0] (* Amiram Eldar, May 05 2023 *)

PROG

(PARI) a(n)=binomial(2*n, n)/(n+1)*round(((1+sqrt(2))^(n+1)-(1-sqrt(2))^(n+1))/(2*sqrt(2)))

CROSSREFS

Cf. A000129, A000108, A098614, A098617, A098618, A000984, A214377.

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Oct 09 2004

STATUS

approved

A098619

G.f. A(x) satisfies: A(x*G098618(x)) = G098618(x), where G098618 is the g.f. for A098618(n) = A007482(n)*Catalan(n).

+10
2

1, 3, 13, 51, 213, 867, 3589, 14739, 60853, 250563, 1033605, 4259571, 17565909, 72412707, 298586661, 1231016019, 5075753589, 20927272323, 86286346693, 355763629491, 1466857936405, 6047981701347, 24936516122469, 102815688922899, 423920292507061, 1747866711689283, 7206641564551429

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

OFFSET

0,2

COMMENTS

G.f. satisfies: A(x) = x/(series reversion of x*G098618(x)), where G098618 is the g.f. for A098618 = {1*1,3*1,11*2,39*5,139*14,495*42,1763*132,...}.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

FORMULA

G.f.: (sqrt(1-8*x^2) + 3*x)/(1-17*x^2).

a(2*n+1) = 3*17^n.

Recurrence: n*a(n) = (25*n-24)*a(n-2) - 136*(n-3)*a(n-4). - Vaclav Kotesovec, Oct 29 2012

MATHEMATICA

Flatten[{1, 3, 13, 51, Table[17^(n/2)*(1/2+1/2*(-1)^n + 3/34*Sqrt[17]*(1-(-1)^n) + Sum[(-1)^j*(4/17 + Sum[Binomial[2*k-1, k-1]*2^(k+3)/ ((k+1)*17^(k+1)), {k, 1, Floor[(j-1)/2]}]), {j, 3, n-1}]), {n, 4, 20}]}] (* Vaclav Kotesovec, Oct 29 2012 *)

PROG

(PARI) a(n)=polcoeff((sqrt(1-8*x^2+x^2*O(x^n))+3*x)/(1-17*x^2), n);

(PARI) x='x+O('x^66); Vec((sqrt(1-8*x^2) + 3*x)/(1-17*x^2)) \\ Joerg Arndt, May 12 2013

CROSSREFS

Cf. A098618, A098615, A098617, A007482, A000108.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 14 2004

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

A194349

E.g.f.: -log( sqrt(1-x^2) - x ).

+10
1

1, 2, 5, 24, 129, 960, 7965, 80640, 903105, 11612160, 163451925, 2554675200, 43259364225, 797058662400, 15764670046125, 334764638208000, 7571150452490625, 182111963185152000, 4634731528895593125, 124564582818643968000

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

OFFSET

1,2

COMMENTS

Compare e.g.f. to arccosh(x) = log(sqrt(x^2-1) + x).

LINKS

Table of n, a(n) for n=1..20.

FORMULA

a(2*n) = 2^n*(2*n-1)! for n>=1.

a(n) = A100097(n+1)*(n-1)!/2^n for n>=1.

a(n) = (n-1)!/2^n * Sum_{k=0..floor((n+1)/2)} C(n+1,k)*A000129(n+1-2*k) for n >= 1. [From a formula of Paul Barry in A100097]

E.g.f.: log( (sqrt(1-x^2) + x)/(1-2*x^2) ).

EXAMPLE

E.g.f.: A(x) = x + 2*x^2/2! + 5*x^3/3! + 24*x^4/4! + 129*x^5/5! + ...

where

exp(A(x)) = 1 + 2*(x/2) + 6*(x/2)^2 + 16*(x/2)^3 + 46*(x/2)^4 + 128*(x/2)^5 + ... + A098617(n)*(x/2)^n + ...

MATHEMATICA

With[{nn=30}, Rest[CoefficientList[Series[-Log[Sqrt[1-x^2]-x], {x, 0, nn}], x] Range[0, nn]!]] (* Harvey P. Dale, Dec 01 2011 *)

PROG

(PARI) {a(n)=n!*polcoeff(-log(sqrt(1-x^2+x*O(x^n))-x), n)}

(PARI) {A000129(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}

{a(n)=if(n<1, 0, sum(k=0, floor((n+1)/2), binomial(n+1, k)*A000129(n+1-2*k))*(n-1)!/2^n)}

CROSSREFS

Cf. A098617, A100097.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 21 2011

STATUS

approved

A201093

Triangle T(n,k), read by rows, given by (0,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,...) DELTA (1,0,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

+10
1

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 3, 1, 0, 0, 2, 3, 4, 1, 0, 2, 2, 4, 6, 5, 1, 0, 0, 4, 6, 8, 10, 6, 1, 0, 5, 5, 9, 13, 15, 15, 7, 1, 0, 0, 10, 15, 20, 25, 26, 21, 8, 1, 0, 14, 14, 24, 34, 41, 45, 42, 28, 9, 1

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

OFFSET

0,9

COMMENTS

Riordan array (1,xf(x)) where f(x) is g.f. of A097331.

Row sums are in A001405.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A001405(n), A098617(n) for x = 0,1,2 respectively.

EXAMPLE

Triangle begins :

0, 1

0, 1, 1

0, 0, 2, 1

0, 1, 1, 3, 1

0, 0, 2, 3, 4, 1

0, 2, 2, 4, 6, 5, 1

0, 0, 4, 6, 8, 10, 6, 1

0, 5, 5, 9, 13, 15, 15, 7, 1

0, 0, 10, 15, 20, 25, 26, 21, 8, 1

0, 14, 14, 24, 34, 41, 45, 42, 28, 9, 1

0, 0, 28, 42, 56, 70, 78, 77, 64, 36, 10, 1

0, 42, 42, 70, 98, 120, 136, 140, 126, 93, 45, 11, 1

CROSSREFS

Cf. Diagonals : A000012, A001477, A000217, A000215,

Columns :A000007, A097331, A089408

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Nov 26 2011

STATUS

approved

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