A126120 Catalan numbers (A000108) interpolated with 0's.

1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132, 0, 429, 0, 1430, 0, 4862, 0, 16796, 0, 58786, 0, 208012, 0, 742900, 0, 2674440, 0, 9694845, 0, 35357670, 0, 129644790, 0, 477638700, 0, 1767263190, 0, 6564120420, 0, 24466267020, 0, 91482563640, 0, 343059613650, 0

Offset

0,5

Comments

Inverse binomial transform of A001006.

The Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...].

Counts returning walks (excursions) of length n on a 1-d integer lattice with step set {+1,-1} which stay in the chamber x >= 0. - Andrew V. Sutherland, Feb 29 2008

Moment sequence of the trace of a random matrix in G=USp(2)=SU(2). If X=tr(A) is a random variable (A distributed according to the Haar measure on G) then a(n) = E[X^n]. - Andrew V. Sutherland, Feb 29 2008

Essentially the same as A097331. - R. J. Mathar, Jun 15 2008

Number of distinct proper binary trees with n nodes. - Chris R. Sims (chris.r.sims(AT)gmail.com), Jun 30 2010

-a(n-1), with a(-1):=0, n>=0, is the Z-sequence for the Riordan array A049310 (Chebyshev S). For the definition see that triangle. - Wolfdieter Lang, Nov 04 2011

See A180874 (also A238390 and A097610) and A263916 for relations to the general Bell A036040, cycle index A036039, and cumulant expansion polynomials A127671 through the Faber polynomials. - Tom Copeland, Jan 26 2016

A signed version is generated by evaluating polynomials in A126216 that are essentially the face polynomials of the associahedra. This entry's sequence is related to an inversion relation on p. 34 of Mizera, related to Feynman diagrams. - Tom Copeland, Dec 09 2019

References

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Ch. 49, Hemisphere Publishing Corp., 1987.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.

Martin Aigner, Catalan and other numbers: a recurrent theme, in Algebraic Combinatorics and Computer Science, a Tribute to Gian-Carlo Rota, pp.347-390, Springer, 2001.

Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).

C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.

Radica Bojicic, Marko D. Petkovic and Paul Barry, Hankel transform of a sequence obtained by series reversion II-aerating transforms, arXiv:1112.1656 [math.CO], 2011.

Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020.

Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv:1110.6638 [math.NT], 2011.

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Kiran S. Kedlaya and Andrew V. Sutherland, HyperellipticCurves, L-Polynomials, and Random Matrices. In: Arithmetic, Geometry, Cryptography, and Coding Theory: International Conference, November 5-9, 2007, CIRM, Marseilles, France. (Contemporary Mathematics; v.487)

S. Mizera, Combinatorics and Topology of Kawai-Lewellen-Tye Relations, arXiv:1706.08527 [hep-th], 2017.

E. Rowland, Pattern avoidance in binary trees, J. Comb. Theory A 117 (6) (2010) 741-758, Sec. 3.1.

Yidong Sun and Fei Ma, Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths, arXiv:1305.2015 [math.CO], 2013.

Y. Wang and Z.-H. Zhang, Combinatorics of Generalized Motzkin Numbers, J. Int. Seq. 18 (2015) # 15.2.4.

Formula

a(2*n) = A000108(n), a(2*n+1) = 0.

a(n) = A053121(n,0).

(1/Pi) Integral_{0 .. Pi} (2*cos(x))^n *2*sin^2(x) dx. - Andrew V. Sutherland, Feb 29 2008

G.f.: (1 - sqrt(1 - 4*x^2)) / (2*x^2) = 1/(1-x^2/(1-x^2/(1-x^2/(1-x^2/(1-...(continued fraction). - Philippe Deléham, Nov 24 2009

G.f. A(x) satisfies A(x) = 1 + x^2*A(x)^2. - Vladimir Kruchinin, Feb 18 2011

E.g.f.: I_1(2x)/x Where I_n(x) is the modified Bessel function. - Benjamin Phillabaum, Mar 07 2011

Apart from the first term the e.g.f. is given by x*HyperGeom([1/2],[3/2,2], x^2). - Benjamin Phillabaum, Mar 07 2011

a(n) = Integral_{x=-2..2} x^n*sqrt((2-x)*(2+x)))/(2*Pi). - Peter Luschny, Sep 11 2011

E.g.f.: E(0)/(1-x) where E(k) = 1-x/(1-x/(x-(k+1)*(k+2)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013

G.f.: 3/2- sqrt(1-4*x^2)/2 = 1/x^2 + R(0)/x^2, where R(k) = 2*k-1 - x^2*(2*k-1)*(2*k+1)/R(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 28 2013 (warning: this is not the g.f. of this sequence, R. J. Mathar, Sep 23 2021)

G.f.: 1/Q(0), where Q(k) = 2*k+1 + x^2*(1-4*(k+1)^2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 09 2014

a(n) = n!*[x^n]hypergeom([],[2],x^2). - Peter Luschny, Jan 31 2015

a(n) = 2^n*hypergeom([3/2,-n],[3],2). - Peter Luschny, Feb 03 2015

a(n) = ((-1)^n+1)*2^(2*floor(n/2)-1)*Gamma(floor(n/2)+1/2)/(sqrt(Pi)* Gamma(floor(n/2)+2)). - Ilya Gutkovskiy, Jul 23 2016

D-finite with recurrence (n+2)*a(n) +4*(-n+1)*a(n-2)=0. - R. J. Mathar, Mar 21 2021

From Peter Bala, Feb 03 2024: (Start)

a(n) = 2^n * Sum_{k = 0..n} (-2)^(-k)*binomial(n, k)*Catalan(k+1).

G.f.: 1/(1 + 2*x) * c(x/(1 + 2*x))^2 = 1/(1 - 2*x) * c(-x/(1 - 2*x))^2 = c(x^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

Example

G.f. = 1 + x^2 + 2*x^4 + 5*x^6 + 14*x^8 + 42*x^10 + 132*x^12 + 429*x^14 + ...
From Gus Wiseman, Nov 14 2022: (Start)
The a(0) = 1 through a(8) = 14 ordered binary rooted trees with n + 1 nodes (ranked by A358375):
  o  .  (oo)  .  ((oo)o)  .  (((oo)o)o)  .  ((((oo)o)o)o)
                 (o(oo))     ((o(oo))o)     (((o(oo))o)o)
                             ((oo)(oo))     (((oo)(oo))o)
                             (o((oo)o))     (((oo)o)(oo))
                             (o(o(oo)))     ((o((oo)o))o)
                                            ((o(o(oo)))o)
                                            ((o(oo))(oo))
                                            ((oo)((oo)o))
                                            ((oo)(o(oo)))
                                            (o(((oo)o)o))
                                            (o((o(oo))o))
                                            (o((oo)(oo)))
                                            (o(o((oo)o)))
                                            (o(o(o(oo))))
(End)

Maple

with(combstruct): grammar := { BB = Sequence(Prod(a, BB, b)), a = Atom, b = Atom }: seq(count([BB, grammar], size=n), n=0..47); # Zerinvary Lajos, Apr 25 2007
BB := {E=Prod(Z, Z), S=Union(Epsilon, Prod(S, S, E))}: ZL:=[S, BB, unlabeled]: seq(count(ZL, size=n), n=0..45); # Zerinvary Lajos, Apr 22 2007
BB := [T, {T=Prod(Z, Z, Z, F, F), F=Sequence(B), B=Prod(F, Z, Z)}, unlabeled]: seq(count(BB, size=n+1), n=0..45); # valid for n> 0. # Zerinvary Lajos, Apr 22 2007
seq(n!*coeff(series(hypergeom([], [2], x^2), x, n+2), x, n), n=0..45); # Peter Luschny, Jan 31 2015
# Using function CompInv from A357588.
CompInv(48, n -> ifelse(irem(n, 2) = 0, 0, (-1)^iquo(n-1, 2))); # Peter Luschny, Oct 07 2022

Mathematica

a[n_?EvenQ] := CatalanNumber[n/2]; a[n_] = 0; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Sep 10 2012 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ BesselI[ 1, 2 x] / x, {x, 0, n}]]; (* Michael Somos, Mar 19 2014 *)
bot[n_]:=If[n==1, {{}}, Join@@Table[Tuples[bot/@c], {c, Table[{k, n-k-1}, {k, n-1}]}]];
Table[Length[bot[n]], {n, 10}] (* Gus Wiseman, Nov 14 2022 *)
Riffle[CatalanNumber[Range[0, 50]], 0, {2, -1, 2}] (* Harvey P. Dale, May 28 2024 *)

Prog

(Sage)
def A126120_list(n) :
    D = [0]*(n+2); D[1] = 1
    b = True; h = 2; R = []
    for i in range(2*n-1) :
        if b :
            for k in range(h, 0, -1) : D[k] -= D[k-1]
            h += 1; R.append(abs(D[1]))
        else :
            for k in range(1, h, 1) : D[k] += D[k+1]
        b = not b
    return R
A126120_list(46) # Peter Luschny, Jun 03 2012
(Magma) &cat [[Catalan(n), 0]: n in [0..30]]; // Vincenzo Librandi, Jul 28 2016
(Python)
from math import comb
def A126120(n): return 0 if n&1 else comb(n, m:=n>>1)//(m+1) # Chai Wah Wu, Apr 22 2024

Crossrefs

Cf. A000108, A036039, A036040, A097610, A127671, A180874, A238390, A263916.

Cf. A126216.

The unordered version is A001190, ranked by A111299.

These trees (ordered binary rooted) are ranked by A358375.

Cf. A000081, A001263, A005043, A032027, A063895, A245824.

Keyword

nonn,easy

Author

Philippe Deléham, Mar 06 2007

Extensions

An erroneous comment removed by Tom Copeland, Jul 23 2016

Status

approved

A097331 Expansion of 1 + 2x/(1 + sqrt(1 - 4x^2)).

1, 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132, 0, 429, 0, 1430, 0, 4862, 0, 16796, 0, 58786, 0, 208012, 0, 742900, 0, 2674440, 0, 9694845, 0, 35357670, 0, 129644790, 0, 477638700, 0, 1767263190, 0, 6564120420, 0, 24466267020, 0, 91482563640, 0, 343059613650, 0

Offset

0,6

Comments

Binomial transform is A097332. Second binomial transform is A014318.

Essentially the same as A126120. - R. J. Mathar, Jun 15 2008

Hankel transform is A087960(n) = (-1)^binomial(n+1,2). - Paul Barry, Aug 10 2009

Links

Michael De Vlieger, Table of n, a(n) for n = 0..3340

Jean-Luc Baril, Sergey Kirgizov, Armen Petrossian, Motzkin paths with a restricted first return decomposition, Integers (2019) Vol. 19, A46.

Formula

a(n) = 0^n + Catalan((n-1)/2)(1-(-1)^n)/2.

Unsigned version of A090192, A105523. - Philippe Deléham, Sep 29 2006

From Paul Barry, Aug 10 2009: (Start)

G.f.: 1+xc(x^2), c(x) the g.f. of A000108;

G.f.: 1/(1-x/(1+x/(1+x/(1-x/(1-x/(1+x/(1+x/(1-x/(1-x/(1+... (continued fraction);

G.f.: 1+x/(1-x^2/(1-x^2/(1-x^2/(1-x^2/(1-... (continued fraction). (End)

G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1+2*x) (continued fraction); more generally g.f. C(x/(1+2*x)) where C(x) is the g.f. for the Catalan numbers (A000108). - Joerg Arndt, Mar 18 2011

Conjecture: (n+1)*a(n) + n*a(n-1) + 4*(-n+2)*a(n-2) + 4*(-n+3)*a(n-3)=0. - R. J. Mathar, Dec 02 2012

Recurrence: (n+3)*a(n+2) = 4*n*a(n), a(0)=a(1)=1. For nonzero terms, a(n) ~ 2^(n+1)/((n+1)^(3/2)*sqrt(2*Pi)). - Fung Lam, Mar 17 2014

Maple

A097331_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w]:=a[w-1]-(-1)^w*add(a[j]*a[w-j-1], j=1..w-1) od; convert(a, list)end: A097331_list(48); # Peter Luschny, May 19 2011

Mathematica

a[0] = 1; a[n_?OddQ] := CatalanNumber[(n-1)/2]; a[_] = 0; Table[a[n], {n, 0, 48}] (* Jean-François Alcover, Jul 24 2013 *)

Prog

(Sage)
def A097331_list(n) :
    D = [0]*(n+2); D[1] = 1
    b = True; h = 1; R = []
    for i in range(2*n-1) :
        if b :
            for k in range(h, 0, -1) : D[k] -= D[k-1]
            h += 1; R.append(abs(D[1]))
        else :
            for k in range(1, h, 1) : D[k] += D[k+1]
        b = not b
    return R
A097331_list(49) # Peter Luschny, Jun 03 2012

Keyword

easy,nonn

Author

Paul Barry, Aug 05 2004

Status

approved

A090192 Carlitz-Riordan q-Catalan numbers (recurrence version) for q = -1.

1, 1, 0, -1, 0, 2, 0, -5, 0, 14, 0, -42, 0, 132, 0, -429, 0, 1430, 0, -4862, 0, 16796, 0, -58786, 0, 208012, 0, -742900, 0, 2674440, 0, -9694845, 0, 35357670, 0, -129644790, 0, 477638700, 0, -1767263190, 0, 6564120420, 0, -24466267020, 0, 91482563640, 0, -343059613650, 0

Offset

0,6

Comments

Hankel transform is (-1)^C(n+1,2). - Paul Barry, Feb 15 2008

Links

Fung Lam and Seiichi Manyama, Table of n, a(n) for n = 0..3338 (first 1002 terms from Fung Lam)

Formula

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-1 and a(0)=1.

G.f.: 1+x*c(-x^2), where c(x) is the g.f. of A000108; a(n) = 0^n+C((n-1)/2)(-1)^((n-1)/2)(1-(-1)^n)/2, where C(n) = A000108(n). - Paul Barry, Feb 15 2008

G.f.: 1/(1-x/(1+x/(1-x/(1+x/(1-x/(1+x/(1-.... (continued fraction). - Paul Barry, Jan 15 2009

a(n) = 2 * a(n-1) - Sum_{k=1..n} a(k-1) * a(n-k) if n>0. - Michael Somos, Jul 23 2011

G.f.: (2*x-1+sqrt(1+4*x^2))/(2*x). - Philippe Deléham, Nov 07 2011

E.g.f.: x*hypergeom([1/2],[2,3/2],-x^2)=A(x)=x*(1-(x^2)/(Q(0)+(x^2)); Q(k)=2*(k^3)+9*(k^2)+(13-2*(x^2))*k-(x^2)+6+(x^2)*(k+1)*(k+2)*((2*k+3)^2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 22 2011

G.f.: 2 + (G(0)-1)/(2*x) where G(k)=1 - 4*x/(1 + 1/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 08 2012

G.f.: 2 + (G(0) -1)/x, where G(k)= 1 - x/(1 + x/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jul 17 2013

G.f.: 1 - 1/(2*x) + G(0)/(4*x), where G(k)= 1 + 1/(1 - 2*x^2*(2*k-1)/(2*x^2*(2*k-1) - (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 17 2013

G.f.: 1- x/(Q(0) + 2*x^2), where Q(k)= (4*x^2 - 1)*k - 2*x^2 - 1 + 2*x^2*(k+1)*(2*k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 17 2013

G.f.: 1+ x/Q(0), where Q(k) = 2*k+1 - x^2*(1-4*(k+1)^2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 09 2014

D-finite with recurrence: (n+3)*a(n+2) = -4*n*a(n), a(0)=a(1)=1. For nonzero terms, a(n) ~ (-1)^((n+3)/2)/sqrt(2*Pi)*2^(n+1)/(n+1)^(3/2). - Fung Lam, Mar 17 2014

a(n) = hypergeom([-n+1,-n], [2], -1). - Peter Luschny, Sep 22 2014

G.f. A(x) satisfies A(x) = 1 / (1 - x * A(-x)). - Michael Somos, Dec 26 2016

From Peter Bala, May 13 2024: (Start)

a(n) = 2^n * Integral_{x = 0..1} LegendreP(n, x) dx.

a(n) = Sum_{k = 0..floor(n/2)} (-1)^k*binomial(n,k)*binomial(2*n-2*k,n)/(n-2*k+1).

a(n) = Sum_{k = 0..n} (-1)^k * 2^(n-k)*binomial(n,k)*binomial(n+k,k)/(k + 1).

a(n) = 2^n * hypergeom([n + 1, -n], [2], 1/2).

a(n) = 1/n * Sum_{k = 0..n} (-1)^k*binomial(n,k)*binomial(n,k+1) for n >= 1.

a(n) = 2^(n-1) * Gamma(1/2)/(Gamma((2-n)/2)*Gamma((n+3)/2)). (End)

Example

G.f. = 1 + x - x^3 + 2*x^5 - 5*x^7 + 14*x^9 - 42*x^11 + 132*x^13 - 429*x^15 + ...

Maple

A090192_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]-add(a[j]*a[w-j-1], j=1..w-1) od;
convert(a, list) end: A090192_list(48); # Peter Luschny, May 19 2011
a := n -> hypergeom([-n+1, -n], [2], -1); seq(round(evalf(a(n), 69)), n=0..48); # Peter Luschny, Sep 22 2014
a:= proc(n) if n::even then 0 else (-1)^((n-1)/2)*binomial(n+1, (n+1)/2)/(2*n) fi end proc: a(0):= 1:
seq(a(n), n=0..100); # Robert Israel, Sep 22 2014

Mathematica

CoefficientList[Series[(2 x - 1 + Sqrt[1 + 4*x^2])/(2 x), {x, 0, 50}],
  x] (* G. C. Greubel, Dec 24 2016 *)
Table[Hypergeometric2F1[1 - n, -n, 2, -1], {n, 0, 48}] (* Michael De Vlieger, Dec 26 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, n++; A = vector(n); A[1] = 1; for( k=2, n, A[k] = 2 * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 23 2011 */
(Sage)
def A090192_list(n) :
    D = [0]*(n+2); D[1] = 1
    b = True; h = 1; R = []
    for i in range(2*n-1) :
        if b :
            for k in range(h, 0, -1) : D[k] -= D[k-1]
            h += 1; R.append(D[1])
        else :
            for k in range(1, h, 1) : D[k] += D[k+1]
        b = not b
    return R
A090192_list(49) # Peter Luschny, Jun 03 2012
(Ruby)
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
  ary
end
def A090192(n)
  A(-1, n)
end # Seiichi Manyama, Dec 24 2016
(PARI) Vec((2*x - 1 + sqrt(1+4*x^2))/(2*x) + O(x^50)) \\ G. C. Greubel, Dec 24 2016

Crossrefs

Cf. A227543.

Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), this sequence (q=-1), A000108 (q=1), A015083 (q=2), A015084 (q=3), A015085 (q=4), A015086 (q=5), A015089 (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).

Column k=1 of A290789.

Keyword

sign

Author

Philippe Deléham, Jan 22 2004

Status

approved

A105523 Expansion of 1-x*c(-x^2) where c(x) is the g.f. of A000108.

1, -1, 0, 1, 0, -2, 0, 5, 0, -14, 0, 42, 0, -132, 0, 429, 0, -1430, 0, 4862, 0, -16796, 0, 58786, 0, -208012, 0, 742900, 0, -2674440, 0, 9694845, 0, -35357670, 0, 129644790, 0, -477638700, 0, 1767263190, 0

Offset

0,6

Comments

Row sums of A105522. Row sums of inverse of A105438.

First column of number triangle A106180.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 313.

R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.

Formula

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

a(n) = 0^n + sin(Pi*(n-2)/2)(C((n-1)/2)(1-(-1)^n)/2).

G.f.: 1/(1+x/(1-x/(1+x/(1-x/(1+x/(1-x.... (continued fraction). - Paul Barry, Jan 15 2009

a(n) = Sum{k = 0..n} A090181(n,k)*(-1)^k. - Philippe Deléham, Feb 02 2009

a(n) = (1/n)*sum_{i = 0..n-1} (-2)^i*binomial(n, i)*binomial(2*n-i-2, n-1). - Vladimir Kruchinin, Dec 26 2010

With offset 1, a(n) = -2 * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k), for n>1. - Michael Somos, Jul 25 2011

D-finite with recurrence: (n+3)*a(n+2) = -4*n*a(n), a(0)=1, a(1)=-1. - Fung Lam, Mar 18 2014

For nonzero terms, a(n) ~ (-1)^((n+1)/2)/sqrt(2*Pi)*2^(n+1)/(n+1)^(3/2). - Fung Lam, Mar 17 2014

a(n) = -(sqrt(Pi)*2^(n-1))/(Gamma(1-n/2)*Gamma((n+3)/2)) for n odd. - Peter Luschny, Oct 31 2014

From Peter Bala, Apr 20 2024: (Start)

a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n + k, 2*k)*Catalan(k), where Catalan(k) = A000108(k).

a(n) = (-2)^n * hypergeom([-n, n+1], [2], 1/2).

O.g.f.: A(x) = 1/x * series reversion of x*(1 - x)/(1 - 2*x). Cf. A152681. (End)

Example

G.f. = 1 - x + x^3 - 2*x^5 + 5*x^7 - 14*x^9 + 42*x^11 - 132*x^13 + 429*x^15 + ...

Maple

A105523_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w]:=-a[w-1]+(-1)^w*add(a[j]*a[w-j-1], j=1..w-1) od; convert(a, list)end: A105523_list(40); # Peter Luschny, May 19 2011

Mathematica

a[n_?EvenQ] := 0; a[n_?OddQ] := 4^n*Gamma[n/2] / (Gamma[-n/2]*(n+1)!); a[0] = 1; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)
CoefficientList[Series[(1 + 2 x - Sqrt[1 + 4 x^2])/(2 x), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 01 2014 *)
a[ n_] := SeriesCoefficient[ (1 + 2 x - Sqrt[ 1 + 4 x^2]) / (2 x), {x, 0, n}]; (* Michael Somos, Jun 17 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], a[n] = -2 a[n - 1] + Sum[ a[j] a[n - j - 1], {j, 0, n - 1}]]; (* Michael Somos, Jun 17 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, n++; A = vector(n); A[1] = 1; for( k=2, n, A[k] = -2 * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */
(Sage)
def A105523(n):
    if is_even(n): return 0 if n>0 else 1
    return -(sqrt(pi)*2^(n-1))/(gamma(1-n/2)*gamma((n+3)/2))
[A105523(n) for n in (0..29)] # Peter Luschny, Oct 31 2014
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 + 2*x - Sqrt(1+4*x^2))/(2*x))); // G. C. Greubel, Sep 16 2018

Crossrefs

Cf. A000108, A097331, A090192, A090181, A105522, A105438.

Keyword

easy,sign

Author

Paul Barry, Apr 11 2005

Extensions

Typo in definition corrected by Robert Israel, Oct 31 2014

Status

approved

A210628 Expansion of (-1 + 2*x + sqrt( 1 - 4*x^2)) / (2*x) in powers of x.

1, -1, 0, -1, 0, -2, 0, -5, 0, -14, 0, -42, 0, -132, 0, -429, 0, -1430, 0, -4862, 0, -16796, 0, -58786, 0, -208012, 0, -742900, 0, -2674440, 0, -9694845, 0, -35357670, 0, -129644790, 0, -477638700, 0, -1767263190, 0, -6564120420, 0, -24466267020, 0

Offset

0,6

Comments

Except for the leading term, the sequence is equal to -A097331(n). - Fung Lam, Mar 22 2014

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

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

G.f. A(x) satisfies 0 = f(x, A(x)) where f(x, y) = x*y^2 - (1 - 2*x) * (1 - y).

G.f. A(x) satisfies A( x / (1 + x^2) ) = 1 - x.

G.f. A(x) = 1 - x - x * (1 - A(x))^2 = 1 - 1/x + 1 / (1 - A(x)).

G.f. A(x) = 1 / (1 + x / (1 - 2*x + x * A(x))).

G.f. A(x) = 1 / (1 + x / (1 - x / (1 - x / (1 + x * A(x))))).

G.f. A(x) = 1 / (1 + x * A001405(x)). A126930(x) = 1 / (1 + x * A(x)).

G.f. A(x) = 1 - x / (1 - x^2 / (1 - x^2 / (1 - x^2 / ...))). - Michael Somos, Jan 02 2013

a(2*n) = 0 unless n=0, a(2*n + 1) = -A000108(n). a(n) = (-1)^n * A097331(n). a(n-1) = (-1)^floor(n/2) * A090192(n).

Convolution inverse of A210736. - Michael Somos, Jan 02 2013

G.f.: 2/( G(0) + 1), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1+2*x) - 2*x*(1+2*x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1+2*x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013

D-finite with recurrence: (n+3)*a(n+2) = 4*n*a(n), a(0)=1, a(1)=-1. - Fung Lam, Mar 17 2014

For nonzero odd-power terms, a(n) = -2^(n+1)/(n+1)^(3/2)/sqrt(2*Pi)*(1+3/(4*n) + O(1/n^2)). (with contribution of Vaclav Kotesovec) - Fung Lam, Mar 17 2014

Example

G.f. = 1 - x - x^3 - 2*x^5 - 5*x^7 - 14*x^9 - 42*x^11 - 132*x^13 - 429*x^15 + ...

Mathematica

CoefficientList[Series[1 - 2 x/(1 + Sqrt[1 - 4 x^2]), {x, 0, 45}], x] (* Bruno Berselli, Mar 25 2012 *)
a[ n_] := SeriesCoefficient[ (-1 + 2 x + Sqrt[1 - 4 x^2]) / (2 x), {x, 0, n}];

Prog

(PARI) {a(n) = polcoeff( (-1 + 2*x + sqrt( 1 - 4*x^2 + x^2 * O(x^n))) / (2*x), n)};
(PARI) {a(n) = if( n<1, n==0, polcoeff( serreverse( -x / (1 + x^2) + x * O(x^n)), n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = 1 - x - x * (1 - A)^2); polcoeff( A, n))};
(Maxima) makelist(coeff(taylor(1-2*x/(1+sqrt(1-4*x^2)), x, 0, n), x, n), n, 0, 45); \\ Bruno Berselli, Mar 25 2012
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((-1 + 2*x + Sqrt(1-4*x^2))/(2*x))); // G. C. Greubel, Aug 11 2018

Crossrefs

Cf. A000108, A001405, A090192, A097331, A126930, A210736.

Keyword

sign

Author

Michael Somos, Mar 25 2012

Status

approved