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Search: a098614 -id:a098614
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A098615
G.f. A(x) satisfies: A(x*G(x)) = G(x), where G(x) is the g.f. for A098614(n) = Fibonacci(n+1)*Catalan(n).
+20
15
1, 1, 3, 5, 13, 25, 61, 125, 295, 625, 1447, 3125, 7151, 15625, 35491, 78125, 176597, 390625, 880125, 1953125, 4390901, 9765625, 21920913, 48828125, 109486993, 244140625, 547018941, 1220703125, 2733608905, 6103515625, 13662695645, 30517578125, 68294088535, 152587890625, 341399727335, 762939453125, 1706739347095, 3814697265625, 8532741458075, 19073486328125, 42660172763995, 95367431640625
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
G.f. satisfies: A(x) = x/(series reversion of x*G098614(x)), where G098614 is the g.f. for A098614 = {1*1, 1*1, 2*2, 3*5, 5*14, 8*42, 13*132, ...}.
Hankel transform is 2^n. Image of F(n+1) under the Riordan array (c(x^2),xc(x^2)), c(x) the g.f. of A000108. The sequence 0,1,1,3,5,... has general term Sum_{k=0..floor(n/2)} (C(n-1,k) - C(n-1,k-1))*F(n-2k). It is the image of the Fibonacci numbers under the transform of generating functions g(x)-> g(xc(x^2)), c(x) the g.f. of A000108. This sequence has Hankel transform -(-4)^((n-1)/2)(1-(-1)^n)/2. - Paul Barry, Oct 01 2007
The sequence of fractions 1, 1/2, 3/4, 5/8, 13/16, 25/32, ... or a(n)/2^n is the image of F(n+1) under the Chebyshev related (rational) Riordan array c((x/2)^2),(x/2)c((x/2)^2)) where c(x) is the g.f. of A000108. The Hankel transform of this fraction sequence is 1/(2^(n^2)). - Paul Barry, Jun 17 2008
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
Paul Barry and A. Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, example 30.
Cyril Banderier, Markus Kuba, and Michael Wallner, Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions, arXiv:2103.03751 [math.PR], 2021.
FORMULA
G.f.: (x + sqrt(1-4*x^2)) / (1-5*x^2).
G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x) + 5*x^2*A(x)^2). - Paul D. Hanna, Nov 18 2014
a(2*n) = A046748(n).
a(2*n+1) = 5^n.
a(n) = Sum_{k=0..floor((n+1)/2)} (C(n,k) - C(n,k-1))*Fibonacci(n-2k+1). - Paul Barry, Oct 01 2007
G.f.: 1/(1-x-2x^2/(1-x^2/(1-x^2/(1-x^2/(1-x^2/(1-.... (continued fraction). - Paul Barry, Feb 09 2009
a(n) = Sum_{k=0..n} binomial((n-1)/2,(n-k)/2)*2^(n-k-1)*(1+(-1)^(n-k)). - Vladimir Kruchinin, Apr 16 2011
From Gary W. Adamson, Sep 22 2011: (Start)
a(n) is the upper left term in M^n, M = an infinite square production matrix as follows:
1, 1, 1, 0, 0, 0, ...
1, 0, 0, 1, 0, 0, ...
1, 0, 0, 0, 1, 0, ...
0, 1, 0, 0, 0, 1, ...
0, 0, 1, 0, 0, 0, ...
0, 0, 0, 1, 0, 0, ...
0, 0, 0, 0, 1, 0, ...
0, 0, 0, 0, 0, 1, ...
... (End)
a(n) = Sum_{k=0..floor(n/2)} A054335(n-k,n-2k). - Philippe Deléham, Feb 01 2012
a(n) = Sum_{k=0..n} A053121(n,k)*A000045(k+1). - Philippe Deléham, Feb 03 2012
n*a(n) +(n-1)*a(n-1) -3*(3*n-4)*a(n-2) -3*(3*n-7)*a(n-3) +20*(n-3)*a(n-4) +20*(n-4)*a(n-5) = 0. - R. J. Mathar, Jul 21 2017
MATHEMATICA
Array[Sum[Binomial[(# - 1)/2, (# - k)/2]*2^(# - k - 1)*((-1)^(# - k) + 1), {k, 0, #}] &, 42, 0] (* or *)
CoefficientList[Series[(Sqrt[1 - 4 x^2] + x)/(1 - 5 x^2), {x, 0, 41}], x] (* Michael De Vlieger, May 20 2021 *)
PROG
(PARI) { a(n) = polcoeff((sqrt(1-4*x^2+x^2*O(x^n))+x)/(1-5*x^2), n) }
for(n=0, 50, print1(a(n), ", "))
(Maxima) a(n):=sum(binomial((n-1)/2, (n-k)/2)*2^(n-k-1)*((-1)^(n-k)+1), k, 0, n); /* Vladimir Kruchinin, Apr 16 2011 */
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (x+Sqrt(1-4*x^2))/(1-5*x^2) )); // G. C. Greubel, Jul 31 2024
(SageMath)
def A098615_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (x+sqrt(1-4*x^2))/(1-5*x^2) ).list()
A098615_list(30) # G. C. Greubel, Jul 31 2024
CROSSREFS
Cf. A000045, A000208, A046748, A053121, A054335, A098614.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 14 2004
STATUS
approved
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
A200375
Product of Catalan and Jacobsthal numbers: a(n) = A000108(n)*A001045(n+1).
+10
4
1, 1, 6, 25, 154, 882, 5676, 36465, 244530, 1657942, 11471668, 80242890, 568080772, 4056976900, 29212908120, 211783889025, 1544811959970, 11328491394990, 83473572128100, 617702666484750, 4588654943721420, 34206312386929020, 255803818897858920, 1918528298674328250, 14427334095935095764
(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), |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
Michael De Vlieger, Table of n, a(n) for n = 0..1112
Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.
Paul Barry and Arnauld Mesinga Mwafise, Classical and Semi-Classical Orthogonal Polynomials Defined by Riordan Arrays, and Their Moment Sequences, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.5.
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-32*x^2))/2 )/(3*x).
G.f.: (1/x)*Series_Reversion(x-x^2 - 4*x^3*Sum_{n>=0} A000108(n)*3^n*x^(2*n) ).
G.f. satisfies: A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) is the g.f. of A200376: G(x) = 1/sqrt(1-10*x^2 + x^4/(1-8*x^2)) + x/(1-9*x^2).
n*(n+1)*a(n) -2*n*(2*n-1)*a(n-1) -8*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 17 2011
a(n) = binomial(2*n,n)/(n+1) * (2^(n+1) + (-1)^n)/3.
From Peter Bala, Aug 17 2021: (Start)
G.f.: A(x) = (sqrt(1 + 4*x) - sqrt(1 - 8*x))/(6*x).
A(x) = 1/sqrt(1 + 4*x)*c( 3*x/(1 + 4*x) ), where c(x) = (1 - sqrt(1- 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. Cf. A151374.
In general, [x^n] ( 1/sqrt(1 + 4*x)*c( k*x/(1 + 4*x) ) ) = Catalan(n)*((k-1)^(n+1) + (-1)^(n+1))/k.
A(x) = 1/sqrt(1 - 8*x)*c( -3*x/(1 - 8*x) ). (End)
EXAMPLE
G.f.: A(x) = 1 + x + 2*3*x^2 + 5*5*x^3 + 14*11*x^4 + 42*21*x^5 + 132*43*x^6 + 429*85*x^7 + 1430*171*x^8 +...+ A000108(n)*A001045(n)*x^n +...
The g.f. of the Jacobsthal sequence A001045, F(x) = 1/(1-x-2*x^2), begins:
F(x) = 1 + x + 3*x^2 + 5*x^3 + 11*x^4 + 21*x^5 + 43*x^6 + 85*x^7 + 171*x^8 +...
The g.f. of A200376, where G(x) = A(x/G(x)), begins:
G(x) = 1 + x + 5*x^2 + 9*x^3 + 37*x^4 + 81*x^5 + 301*x^6 + 729*x^7 +...
in which the odd-indexed coefficients are powers of 9.
MATHEMATICA
Array[CatalanNumber[# - 1] (2^# - (-1)^#)/3 &, 25] (* Michael De Vlieger, Apr 24 2018 *)
PROG
(PARI) {a(n)=binomial(2*n, n)/(n+1)*(2^(n+1)+(-1)^n)/3}
(PARI) {a(n)=polcoeff(sqrt((1-2*x - sqrt(1-4*x-32*x^2+O(x^(n+3))))/2)/(3*x), n)}
(PARI) {a(n)=polcoeff((1/x)*serreverse(x-x^2 - 4*x^3*sum(m=0, n\2, binomial(2*m, m)/(m+1)*3^m*x^(2*m))+x^3*O(x^n)), n)}
CROSSREFS
Cf. A200376, A098614, A098616, A200312.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 16 2011
EXTENSIONS
Typo in Name corrected by Peter Bala, Aug 17 2021
STATUS
approved
A098618
Products of A007482 and Catalan numbers: a(n) = A007482(n)*A000108(n).
+10
3
1, 3, 22, 195, 1946, 20790, 232716, 2693691, 31979090, 387243714, 4764470932, 59391201870, 748472730628, 9520446996300, 122067269204760, 1575965219205195, 20470515781159170, 267325017886787850
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Radius of convergence: r = (sqrt(17)-3)/16; A(r) = sqrt(2+6/sqrt(17)). Recurrence of A007482 is A007482(n) = 3*A007482(n-1) + 2*A007482(n-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
Table of n, a(n) for n=0..17.
FORMULA
G.f.: A(x) = sqrt((1-6*x - sqrt(1-12*x-32*x^2))/34 )/x.
n*(n+1)*a(n) -6*n*(2*n-1)*a(n-1) -8*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 17 2018
EXAMPLE
Begins: {1*1, 3*1, 11*2, 39*5, 139*14, 495*42, 1763*132, 6279*429,...}.
PROG
(PARI) {a(n)=binomial(2*n, n)/(n+1)*((3+sqrt(17))^(n+1)-(3-sqrt(17))^(n+1))/2^(n+1)/sqrt(17)}
CROSSREFS
Cf. A007482, A000108, A098614, A098616, A098619.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 09 2004
STATUS
approved
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
A200539
Product of Fibonacci and Motzkin numbers: a(n) = A000045(n+1)*A001006(n).
+10
2
1, 1, 4, 12, 45, 168, 663, 2667, 10982, 45925, 194732, 834912, 3614063, 15771795, 69316740, 306534564, 1362986799, 6089916936, 27328613142, 123118156260, 556626199974, 2524659817449, 11484671681511, 52384730922720, 239534402969925, 1097805759803893, 5042014405418968
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The g.f. for the Fibonacci numbers is 1/(1-x-x^2) and the g.f. M(x) for the Motzkin numbers satisfies: M(x) = 1 + x*M(x) + x^2*M(x)^2.
LINKS
Table of n, a(n) for n=0..26.
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 12*x^3 + 45*x^4 + 168*x^5 + 663*x^6 +...
where A(x) = 1*1 + 1*1*x + 2*2*x^2 + 3*4*x^3 + 5*9*x^4 + 8*21*x^5 + 13*51*x^6 + 21*127*x^7 + 34*323*x^8 +...+ A000045(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)}
{a(n)=fibonacci(n+1)*A001006(n)}
CROSSREFS
Cf. A098614, A200538, A200540, A098616.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 18 2011
STATUS
approved
A200540
Product of Pell and Motzkin numbers: a(n) = A000129(n+1)*A001006(n).
+10
2
1, 2, 10, 48, 261, 1470, 8619, 51816, 318155, 1985630, 12561308, 80360280, 519013571, 3379514970, 22161470850, 146227235904, 970126550763, 6467496499590, 43304243215638, 291087137589552, 1963598081845335, 13288619577124122, 90195242361688863, 613843707553183800
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The g.f. for the Pell numbers is 1/(1-2*x-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..23.
EXAMPLE
G.f.: A(x) = 1 + 2*x + 10*x^2 + 48*x^3 + 261*x^4 + 1470*x^5 + 8619*x^6 +...
where A(x) = 1*1 + 2*1*x + 5*2*x^2 + 12*4*x^3 + 29*9*x^4 + 70*21*x^5 + 169*51*x^6 + 408*127*x^7 + 985*323*x^8 +...+ A000129(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)}
{A000129(n)=polcoeff( x/(1-2*x-x^2+x*O(x^n)), n)}
{a(n)=A000129(n+1)*A001006(n)}
CROSSREFS
Cf. A098616, A200538, A200539, A098614.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 18 2011
STATUS
approved
A215931
Product of Fibonacci and Catalan numbers: a(n) = A000045(2*n+2)*A000108(n).
+10
2
1, 3, 16, 105, 770, 6048, 49764, 423423, 3695120, 32891430, 297473956, 2725789248, 25251200716, 236101791900, 2225241057600, 21118368117105, 201640796593290, 1935642349666080, 18670022226540300, 180851385211254450, 1758621701183524320, 17160853351737885660
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
More generally, given {S} such that: S(n) = b*S(n-1) + c*S(n-2) with 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
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: sqrt( (1-6*x - sqrt(1-12*x+16*x^2))/10 )/x.
a(n) = Fibonacci(2*n+2) * binomial(2*n,n)/(n+1).
a(n) = Fibonacci(n+1) * Lucas(n+1) * binomial(2*n,n)/(n+1), where Lucas(n+1) = 2*Fibonacci(n) + Fibonacci(n+1) = A000032(n+1).
a(n) = A000032(n+1) * A098614(n).
n*(n+1)*a(n) -6*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 = 4*sqrt(1-2/sqrt(5)). - Amiram Eldar, May 06 2023
EXAMPLE
G.f.: A(x) = 1 + 3*x + 16*x^2 + 105*x^3 + 770*x^4 + 6048*x^5 + 49764*x^6 +...
such that the coefficients equal the term-wise products:
A = [1*1, 3*1, 8*2, 21*5, 55*14, 144*42, 377*132, 987*429, 2584*1430, ...].
Related expansions.
A(x)^2 = 1 + 6*x + 41*x^2 + 306*x^3 + 2426*x^4 + 20076*x^5 + 171481*x^6 +...
A(x)^3 = 1 + 9*x + 75*x^2 + 630*x^3 + 5400*x^4 + 47223*x^5 + 420277*x^6 +...
Incidentally, note that (2*n+1) divides [x^n] A(x)^3:
A^3 = [1*1, 3*3, 5*15, 7*90, 9*600, 11*4293, 13*32329, 15*253110, ...].
MATHEMATICA
Table[Fibonacci[2*n+2]*Binomial[2*n, n]/(n+1), {n, 0, 25}] (* Vincenzo Librandi, Aug 28 2012 *)
PROG
(PARI) {a(n)=fibonacci(2*n+2)*binomial(2*n, n)/(n+1)}
(PARI) {a(n)=fibonacci(n+1)*(2*fibonacci(n)+fibonacci(n+1))*binomial(2*n, n)/(n+1)}
(PARI) {a(n)=polcoeff( sqrt( (1-6*x - sqrt(1-12*x+16*x^2 +x^3*O(x^n)))/10 )/x, n)}
for(n=0, 21, print1(a(n), ", "))
(Magma) [Fibonacci(2*n+2)*Binomial(2*n, n)/(n+1): n in [0..22]] // Vincenzo Librandi, Aug 28 2012
CROSSREFS
Cf. A098614, A098616, A001906, A000045, A000032.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 27 2012
STATUS
approved
A249925
G.f. satisfies: A(x) = 1 + 2*x*A(x) + 5*x^2*A(x)^2.
+10
2
1, 2, 9, 38, 186, 932, 4889, 26238, 143966, 802652, 4536874, 25932348, 149650516, 870675912, 5101656889, 30078478318, 178309845686, 1062198928812, 6355149937934, 38172142221748, 230094601968876, 1391444403490552, 8439240940653834, 51323083138005388, 312896262064813036, 1911980839096481432
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
FORMULA
G.f.: (1-2*x - sqrt(1-4*x-16*x^2)) / (10*x^2).
Self-convolution square of A098614, where A098614(n) = A000045(n+1)*A000108(n), the term-wise product of Fibonacci and Catalan numbers.
a(n) = Sum_{k=0..n} A000045(k+1)*A000045(n-k+1) * A000108(k)*A000108(n-k).
a(n) = Sum_{k=0..n} Fibonacci(n-k+1)*Fibonacci(k+1) * C(2*(n-k),n-k)*C(2*k,k) / ((n-k+1)*(k+1)).
a(n) == 1 (mod 2) iff n = 2*(2^k - 1) for k>=0.
Given series bisections B0 and B1 such that A(x) = B0(x^2) + x*B1(x^2), then B1(x)/B0(x) = 2 + 10*x*B1(x), thus B1(x) = 2*B0(x)/(1 - 10*x*B0(x)).
a(n) ~ sqrt(5+2*sqrt(5)) * 2^(n+2) * (1+sqrt(5))^n / (5 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 29 2014. Equivalently, a(n) ~ 5^(1/4) * 2^(2*n+2) * phi^(n + 3/2) / (5 * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021
Recurrence: (n+2)*a(n) = 2*(2*n+1)*a(n-1) + 16*(n-1)*a(n-2). - Vaclav Kotesovec, Nov 29 2014
EXAMPLE
G.f.: A(x) = 1 + 2*x + 9*x^2 + 38*x^3 + 186*x^4 + 932*x^5 + 4889*x^6 +...
where the square-root of the g.f. yields
sqrt(A(x)) = 1 + x + 4*x^2 + 15*x^3 + 70*x^4 + 336*x^5 + 1716*x^6 + 9009*x^7 + 48620*x^8 +...+ Fibonacci(n+1)*A000108(n)*x^n + +...
Related expansions.
A(x)^2 = 1 + 4*x + 22*x^2 + 112*x^3 + 605*x^4 + 3292*x^5 + 18298*x^6 +...
which obeys A(x) = 1 + 2*x*A(x) + 5*x^2*A(x)^2.
Given series bisections A(x) = B0(x^2) + x*B1(x^2),
B0(x) = 1 + 9*x + 186*x^2 + 4889*x^3 + 143966*x^4 + 4536874*x^5 +...
B1(x) = 2 + 38*x + 932*x^2 + 26238*x^3 + 802652*x^4 + 25932348*x^5 +...
then B1(x)/B0(x) = 2 + 10*x*B1(x):
B1(x)/B0(x) = 2 + 20*x + 380*x^2 + 9320*x^3 + 262380*x^4 + 8026520*x^5 +...
MATHEMATICA
CoefficientList[Series[(1-2*x-Sqrt[1-4*x-16*x^2]) / (10*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 29 2014 *)
PROG
(PARI) {a(n)=local(X=x+O(x^(n+3)), A); A = (1-2*x - sqrt(1-4*X-16*x^2)) / (10*x^2); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, fibonacci(n-k+1)*fibonacci(k+1)*binomial(2*(n-k), n-k)*binomial(2*k, k)/((n-k+1)*(k+1)))}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A098614, A000045, A000108.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 22 2014
STATUS
approved
A216541
Product of Lucas and Catalan numbers: a(n) = A000032(n+1)*A000108(n).
+10
1
1, 3, 8, 35, 154, 756, 3828, 20163, 108680, 598026, 3342404, 18929092, 108374252, 626264700, 3647936160, 21396522915, 126262239570, 749087596620, 4465444206300, 26733390275130, 160663411399920, 968937572793060, 5862111195487560, 35569106862459300, 216395609659221564
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..24.
FORMULA
G.f.: (1 - sqrt( (1-2*x + sqrt(1-4*x-16*x^2))/2 )) / x.
G.f. satisfies: A(x) = (2+5*x) - (1+4*x)*A(x) + x*(5+2*x)*A(x)^2 - 4*x^2*A(x)^3 + x^3*A(x)^4.
n*(n+1)*a(n) -2*n*(2n-1)*a(n-1) -4*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Sep 11 2012
Sum_{n>=0} a(n)/8^n = 8 - 2*sqrt(10). - Amiram Eldar, May 05 2023
EXAMPLE
G.f.: A(x) = 1 + 3*x + 8*x^2 + 35*x^3 + 154*x^4 + 756*x^5 + 3828*x^6 +...
such that the coefficients equal the term-wise products:
A = [1*1, 3*1, 4*2, 7*5, 11*14, 18*42, 29*132, 47*429, 76*1430, ...].
MATHEMATICA
a[n_] := LucasL[n+1] * CatalanNumber[n]; Array[a, 25, 0] (* Amiram Eldar, May 05 2023 *)
PROG
(PARI) {a(n)=(2*fibonacci(n)+fibonacci(n+1))*binomial(2*n, n)/(n+1)}
(PARI) {a(n)=polcoeff( (1 - sqrt( (1-2*x + sqrt(1-4*x-16*x^2 +x^2*O(x^n)))/2 )) / x, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
Cf. A098614, A215931, A098616, A000032, A000108.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 08 2012
STATUS
approved
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