A106566 -id:A106566 - cp's OEIS frontend

A064310 Generalized Catalan numbers C(-1; n).

1, 1, 0, 1, -2, 6, -18, 57, -186, 622, -2120, 7338, -25724, 91144, -325878, 1174281, -4260282, 15548694, -57048048, 210295326, -778483932, 2892818244, -10786724388, 40347919626, -151355847012, 569274150156

Offset: 0

Wolfdieter Lang, Sep 21 2001

See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.

Unsigned sequence with a(0) := 0 is A000957 (Fine).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. [_N. J. A. Sloane_, Oct 08 2012]
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).
P. Pagacz and M. Wojtylak, On the spectral properties of a class of H-selfadjoint random matrices and the underlying combinatorics, arXiv:1310.2122 [math.PR], 2013.

Cf. A000108, A000957, A039599, A106566.

[1] cat [(1 +(&+[(-2)^k*Binomial(2*k,k)/(k+1): k in [0..n-1]]))/2^n: n in [1..30]]; // G. C. Greubel, Feb 27 2019

a[n_]:= (1/2)^n*(1 + Sum[ CatalanNumber[k]*(-2)^k, {k, 0, n-1}]); Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 17 2013 *)

{a(n) = (1 + sum(k=0, n-1, (-2)^k*binomial(2*k,k)/(k+1)))/2^n};
vector(30, n, n--; a(n)) \\ G. C. Greubel, Feb 27 2019

from itertools import count, islice
def A064310_gen(): # generator of terms
    yield from (1,1,0)
    a, c = 0, 1
    for n in count(1):
        yield (a:=(c:=c*((n<<2)+2)//(n+2))-a>>1)*(1 if n&1 else -1)
A064310_list = list(islice(A064310_gen(),20)) # Chai Wah Wu, Apr 27 2023

[1] + [(1 +sum((-2)^k*catalan_number(k) for k in (0..n-1)))/2^n for n in (1..30)] # G. C. Greubel, Feb 27 2019

a(n) = Sum_{m=0..n-1} (-1)^m*(n-m)*binomial(n-1+m, m)/n.
a(n) = ((1/2)^n)*(1 + Sum_{k=0..n-1} C(k)*(-2)^k ), n >= 1, a(0)= 1, with C(n)=A000108(n) (Catalan).
G.f.: (1+x*c(-x)/2)/(1-x/2) = 1/(1-x*c(-x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{k=0..n} (-1)^(n-k)*A106566(n, k). - Philippe Deléham, Sep 18 2005
(-1)^n*a(n) = Sum_{k=0..n} A039599(n,k)*(-2)^k. - Philippe Deléham, Mar 13 2007
Conjecture: 2*n*a(n) + (7*n-12)*a(n-1) + 2*(-2*n+3)*a(n-2) = 0. - R. J. Mathar, Dec 02 2012

A098474 Triangle read by rows, T(n,k) = C(n,k)C(2k,k)/(k+1), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 12, 20, 14, 1, 5, 20, 50, 70, 42, 1, 6, 30, 100, 210, 252, 132, 1, 7, 42, 175, 490, 882, 924, 429, 1, 8, 56, 280, 980, 2352, 3696, 3432, 1430, 1, 9, 72, 420, 1764, 5292, 11088, 15444, 12870, 4862, 1, 10, 90, 600, 2940, 10584, 27720

Offset: 0

Paul Barry, Sep 09 2004

A Catalan scaled binomial matrix.
From Philippe Deléham, Sep 01 2005: (Start)
Table U(n,k), k >= 0, n >= 0, read by antidiagonals, begins:
  row k = 0: 1, 1,  2,   5,   14, ... is A000108
  row k = 1: 1, 2,  6,  20,   70, ... is A000984
  row k = 2: 1, 3, 12,  50,  280, ... is A007854
  row k = 3: 1, 4, 20, 104,  548, ... is A076035
  row k = 4: 1, 5, 30, 185, 1150, ... is A076036
G.f. for row k: 1/(1-(k+1)*x*C(x)) where C(x) is the g.f. = for Catalan numbers A000108.
U(n,k) = Sum_{j=0..n} A106566(n,j)*(k+1)^j. (End)
This sequence gives the coefficients (increasing powers of x) of the Jensen polynomials for the Catalan sequence A000108 of degree n and shift 0. For the definition of Jensen polynomials for a sequence see a comment in A094436. - Wolfdieter Lang, Jun 25 2019

			Rows begin:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  6,   5;
  1, 4, 12,  20,  14;
  1, 5, 20,  50,  70,  42;
  1, 6, 30, 100, 210, 252, 132;
  ...
Row 3: t*(1 - 3*t + 6*t^2 - 5*t^3)/(1 - 4*t)^(9/2) = 1/2*Sum_{k >= 1} k*(k+1)*(k+2)*(k+3)/4!*binomial(2*k,k)*t^k. - _Peter Bala_, Jun 13 2016

Indranil Ghosh, Rows 0..125, flattened

Row sums are A007317.
Antidiagonal sums are A090344.
Principal diagonal is A000108.
Mirror image of A124644.
Cf. A000984, A007854, A011973, A076035, A076036, A098443, A098473, A106566, A267633.

p := proc(n) option remember; if n = 0 then 1 else normal((x*(1 + 4*x)*diff(p(n-1, x), x) + (2*x + n + 1)*p(n-1, x))/(n + 1)) fi end:
row := n -> local k; seq(coeff(p(n), x, k), k = 0..n):
for n from 0 to 6 do row(n) od;  # Peter Luschny, Jun 21 2023

Table[Binomial[n, k] Binomial[2 k, k]/(k + 1), {n, 0, 10}, {k, 0, n}] // Flatten (* or *)
Table[(-1)^k*CatalanNumber[k] Pochhammer[-n, k]/k!, {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

from functools import cache
@cache
def A098474row(n: int) -> list[int]:
    if n == 0: return [1]
    a = A098474row(n - 1) + [0]
    row = [0] * (n + 1)
    row[0] = 1; row[1] = n
    for k in range(2, n + 1):
        row[k] = (a[k] * (n + k + 1) + a[k - 1] * (4 * k - 2)) // (n + 1)
    return row  # Peter Luschny, Jun 22 2023

def A098474(n,k):
    return (-1)^k*catalan_number(k)*rising_factorial(-n,k)/factorial(k)
for n in range(7): [A098474(n,k) for k in (0..n)] # Peter Luschny, Feb 05 2015

G.f.: 2/(1-x+(1-x-4*x*y)^(1/2)). - Vladeta Jovovic, Sep 11 2004
E.g.f.: exp(x*(1+2*y))*(BesselI(0, 2*x*y)-BesselI(1, 2*x*y)). - Vladeta Jovovic, Sep 11 2004
G.f.: 1/(1-x-xy/(1-xy/(1-x-xy/(1-xy/(1-x-xy/(1-xy/(1-x-xy/(1-xy/(1-... (continued fraction). - Paul Barry, Feb 11 2009
Sum_{k=0..n} T(n,k)*x^(n-k) = A126930(n), A005043(n), A000108(n), A007317(n+1), A064613(n), A104455(n) for x = -2, -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Dec 12 2009
T(n,k) = (-1)^k*Catalan(k)*Pochhammer(-n,k)/k!. - Peter Luschny, Feb 05 2015
O.g.f.: [1 - sqrt(1-4tx/(1-x))]/(2tx) = 1 + (1+t) x + (1+2t+2t^2) x^2 + (1+3t+6t^2+5t^3) x^3 + ... , generating the polynomials of this entry, reverse of A124644. See A011973 for a derivation and the inverse o.g.f., connected to the Fibonacci, Chebyshev, and Motzkin polynomials. See also A267633. - Tom Copeland, Jan 25 2016
From Peter Bala, Jun 13 2016: (Start)
The o.g.f. F(x,t) = ( 1 - sqrt(1 - 4*t*x/(1 - x)) )/(2*t*x) satisfies the partial differential equation d/dx(x*(1 - x)*F) - x*t*(1 + 4*t)*dF/dt - 2*x*t*F = 1. This gives a recurrence for the row polynomials: (n + 2)*R(n+1,t) = t*(1 + 4*t)*R'(n,t) + (2*t + n + 2)*R(n,t), where the prime ' indicates differentiation with respect to t.
Equivalently, setting Q(n,t) = t^(n+2)*R(n,-t)/(1 - 4*t)^(n + 3/2) we have t^2*d/dt(Q(n,t)) = (n + 2)*Q(n+1,t).
This leads to the following expansions:
Q(0,t) = (1/2)*Sum_{k >= 1} k*binomial(2*k,k)*t^(k+1)
Q(1,t) = (1/2)*Sum_{k >= 1} k*(k+1)/2!*binomial(2*k,k)*t^(k+2)
Q(2,t) = (1/2)*Sum_{k >= 1} k*(k+1)*(k+2)/3!*binomial(2*k,k) *t^(k+3) and so on. (End)
Sum_{k=0..n} T(n,k)*x^k = A007317(n+1), A162326(n+1), A337167(n) for x = 1, 2, 3 respectively. - Sergii Voloshyn, Mar 31 2022

New name using a formula of Paul Barry by Peter Luschny, Feb 05 2015

A026726 a(n) = T(2n,n), T given by A026725.

Original entry on oeis.org

1, 2, 7, 27, 108, 440, 1812, 7514, 31307, 130883, 548547, 2303413, 9686617, 40783083, 171868037, 724837891, 3058850316, 12915186640, 54554594416, 230526280814, 974414815782, 4119854160332, 17422801069670, 73695109608352, 311768697325788, 1319136935150530

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Cf. A000045, A000108, A026725.

List([0..30], n-> Sum([0..n], k-> (2*k+1)*Binomial(2*n,n-k)*
Fibonacci(k+1)/(n+k+1) )); # G. C. Greubel, Jul 16 2019

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 4*x*(1-Sqrt(1-4*x))/(8*x^2-(1-Sqrt(1-4*x))^3) )); // G. C. Greubel, Jul 16 2019

A026726 := proc(n)
    A026725(2*n,n) ;
end proc:
seq(A026726(n),n=0..10) ; # R. J. Mathar, Oct 26 2019

CoefficientList[Series[4*x*(1-Sqrt[1-4*x])/(8*x^2-(1-Sqrt[1-4*x])^3), {x,0,30}], x] (* G. C. Greubel, Jul 16 2019 *)

my(x='x+O('x^30)); Vec(4*x*(1-sqrt(1-4*x))/(8*x^2-(1-sqrt(1-4*x))^3)) \\ G. C. Greubel, Jul 16 2019

(4*x*(1-sqrt(1-4*x))/(8*x^2-(1-sqrt(1-4*x))^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 16 2019

From Philippe Deléham, Feb 11 2009: (Start)
a(n) = Sum_{k=0..n} A039599(n,k)*A000045(k+1).
a(n) = Sum_{k=0..n} A106566(n,k)*A122367(k). (End)
From Philippe Deléham, Feb 02 2014: (Start)
a(n) = Sum_{k=0..n} A236843(n+k,2*k).
a(n) = Sum_{k=0..n} A236830(n,k).
a(n) = A236830(n+1,1).
a(n) = A165407(3*n).
G.f.: C(x)/(1-x*C(x)^3), C(x) the g.f. of A000108. (End)
n*(5*n-11)*a(n) +2*(-20*n^2+59*n-30)*a(n-1) +15*(5*n^2-19*n+16)*a(n-2) +2*(5*n-6)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Oct 26 2019
n*a(n) +(-7*n+4)*a(n-1) +(7*n-2)*a(n-2) +(19*n-60)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Oct 26 2019

A076036 G.f.: 1/(1 - 5xC(x)), where C(x) = (1 - sqrt(1 - 4x))/(2x) = g.f. for the Catalan numbers A000108.

Original entry on oeis.org

1, 5, 30, 185, 1150, 7170, 44760, 279585, 1746870, 10916150, 68219860, 426353130, 2664633580, 16653699860, 104084695500, 650526003825, 4065775405350, 25411052086350, 158818913483700, 992617612224750, 6203857867325700, 38774103465635100, 242338116077385600

Offset: 0

N. J. A. Sloane, Oct 29 2002

Numbers have the same parity as the Catalan numbers, that is, a(n) is even except for n of the form 2^m - 1. Follows from C(x) = 1/(1 - x*C(x)) = 1/(1 - 5*x*C(x)) (mod 2). - Peter Bala, Jul 24 2016

Cf. A000108, A000984, A007854, A076035, A126694.

C(x) = (1 - sqrt(1 - 4*x))/(2*x);
my(x = 'x + O('x^25)); Vec(1/(1 - 5*x*C(x))) \\ Michel Marcus, Jan 21 2020

a(n) = Sum_{k = 0..n} A106566(n, k)*5^k. - Philippe Deléham, Sep 01 2005
a(n) = Sum{k = 0..n} A039599(n,k)*4^k. - Philippe Deléham, Sep 08 2007
a(0) = 1, a(n) = (25*a(n-1) - 5*A000108(n-1))/4 for n >= 1. - Philippe Deléham, Nov 27 2007
a(n) = Sum_{k = 0..n} A116395(n,k)*3^k. - Philippe Deléham, Sep 27 2009
D-finite with recurrence: +4*n*a(n) +(-41*n+24)*a(n-1) +50*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020
a(n) = 5*A076025(n), n>0. - R. J. Mathar, Jan 20 2020

A109262 A Catalan transform of the Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 6, 19, 63, 215, 749, 2650, 9490, 34318, 125104, 459152, 1694914, 6287896, 23429158, 87635243, 328917615, 1238303243, 4674847097, 17692789741, 67114622451, 255120892105, 971649360211, 3707176155659, 14167390221873

Offset: 0

Paul Barry, Jun 24 2005

A column of A109267.

Hankel transform is -Fibonacci(2*n). a(n+1) has Hankel transform Fibonacci(2*n+1). - Paul Barry, Nov 22 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
Stoyan Dimitrov, On permutation patterns with constrained gap sizes, arXiv:2002.12322 [math.CO], 2020.
Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
Merve Taştan and Engin Özkan, Catalan transform of the k-Jacobsthal sequence, Electronic Journal of Mathematical Analysis and Applications (2020) Vol. 8, No. 2, 70-74.

Cf. A000045, A000108, A039599, A081696, A106566, A109267.

[n eq 0 select 0 else (&+[k*Binomial(2*n-k-1,n-1)*Fibonacci(k): k in [0..n]])/n: n in [0..30]]; // G. C. Greubel, May 30 2022

CoefficientList[Series[(1-Sqrt[1-4*x])/(2*(Sqrt[1-4*x]+x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

[0]+[(1/n)*sum(k*binomial(2*n-k-1, n-1)*fibonacci(k) for k in (1..n)) for n in (1..30)] # G. C. Greubel, May 30 2022

G.f.: x*c(x)/(1 - x*c(x) - x^2*c(x)^2) = (1 - sqrt(1-4*x))/(2*(x + sqrt(1-4*x))) where c(x) is the g.f. of A000108.
a(n) = Sum_{k=0..n} (k/(2*n-k))*binomial(2*n-k, n-k)*Fibonacci(k).
a(n) = Sum_{k=0..n} A106566(n,k)*A000045(k). - Philippe Deléham, Oct 28 2008
a(n) = Sum_{k=0..n} A039599(n,k)*(-1)^(k+1)*A000045(k). - Philippe Deléham, Oct 28 2008
n*a(n) - (7*n-4)*a(n-1) + (7*n-2)*a(n-2) + (19*n-60)*a(n-3) + 2*(2*n-7)*a(n-4) = 0. - R. J. Mathar, Nov 26 2012
Recurrence: n*(5*n-11)*a(n) = 2*(20*n^2 - 59*n + 30)*a(n-1) - 15*(5*n^2 - 19*n + 16)*a(n-2) - 2*(2*n-5)*(5*n-6)*a(n-3). - Vaclav Kotesovec, Feb 13 2014
a(n) ~ 5*4^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 13 2014
a(n) = (1/(2*sqrt(5)))*Catalan(n-1)*Sum_{j=0..1} ((-1)^j + sqrt(5)) * Hypergeometric2F1([2,1-n], [2*(1-n)], (1+(-1)^j*sqrt(5))/2). - G. C. Greubel, May 30 2022

A121988 Number of vertices of the n-th multiplihedron.

Original entry on oeis.org

0, 1, 2, 6, 21, 80, 322, 1348, 5814, 25674, 115566, 528528, 2449746, 11485068, 54377288, 259663576, 1249249981, 6049846848, 29469261934, 144293491564, 709806846980, 3506278661820, 17385618278700, 86500622296800, 431718990188850, 2160826237261692

Offset: 0

Jonathan Vos Post, Jun 24 2007

G.f. = x*c(x)*c(x*c(x)) where c(x) is the generating function of the Catalan numbers C(n). Thus a(n) is the Catalan transform of the sequence C(n-1). Reference for the definition of Catalan transform is the paper by Paul Barry. - Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007
A129442 is an essentially identical sequence. - R. J. Mathar, Jun 13 2008
From Peter Bala, Jan 27 2020: (Start)
This sequence is the main diagonal of the lower triangular array formed by putting the sequence [0, 1, 1, 2, 5, 14, 42, ...] of Catalan numbers (with 0 prepended) in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.
    0
    1    1
    1    2    2
    2    4    6    6
    5    9   15   21   21
   14   23   38   59   80   80
  ...
Cf. A307495.
Alternatively, the sequence can be obtained by multiplying the sequence of Catalan numbers by the array A106566. (End)

			G.f. = x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 322*x^6 + 1348*x^7 + 5814*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..500
R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO], 2004. See p. 19.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5, pp. 1-24.
David Callan, A combinatorial interpretation of the Catalan transform of the Catalan numbers, arXiv:1111.0996 [math.CO], 2011.
Stefan Forcey, Convex Hull Realizations of the Multiplihedra, Theorem 3.2, p. 8, arXiv:0706.3226 [math.AT], 2007-2008.
Stefan Forcey, Aaron Lauve, and Frank Sottile, New Hopf Structures on Binary Trees, dmtcs:2740 - Discrete Mathematics & Theoretical Computer Science, January 1, 2009, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009).
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Tian-Xiao He and Louis W. Shapiro, Row sums and alternating sums of Riordan arrays, Linear Algebra and its Applications, Volume 507, 15 October 2016, Pages 77-95.

Cf. A000108, A127632, A129442, A007317, A164965, A106566, A158826, A307495.

a:= proc(n) option remember; `if`(n<3, n, (14*(n-1)*(2*n-3)*a(n-1)
      -4*(4*n-9)*(4*n-7)*a(n-2))/ (3*n*(n-1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 20 2012

a[0] = 0; a[n_] := a[n] = (2 n - 2)!/((n - 1)! n!) + Sum[ a[i]*a[n - i], {i, n - 1}]; Table[ a@n, {n, 0, 24}] (* Robert G. Wilson v, Jun 28 2007 *)
a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ x - 2 x^2 + 2 x^3 - x^4, {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, Jun 01 2014 *)
a[0] = 0; a[n_] := Binomial[2n-2, n-1]*Hypergeometric2F1[1/2, 1-n, 2-2n, 4] /n; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 31 2016 *)

{a(n) = if( n<1, 0, polcoeff( serreverse( x - 2*x^2 + 2*x^3 - x^4 + x * O(x^n)), n))}; /* Michael Somos, Jun 01 2014 */

a(0) = 0; a(n) = C(n-1) + Sum_{i=1..n-1} a(i)*a(n-i), where C(n) = A000108(n).
G.f.: (1-sqrt(2*sqrt(1-4x)-1))/2. a(n) = (1/n)*Sum_{k=1..n} binomial(2*n-k-1,n-1)*binomial(2k-2, k-1); a(0)=0. - Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007
a(n) = Sum_{k = 0..n} A106566(n,k)*A000108(k-1) with A000108(-1)=0. - Philippe Deléham, Aug 27 2007
From Vaclav Kotesovec, Oct 19 2012: (Start)
D-finite with recurrence 3*(n-1)*n*a(n) = 14*(n-1)*(2*n-3)*a(n-1) - 4*(4*n-9)*(4*n-7)*a(n-2).
a(n) ~ 2^(4*n-5/2)/(sqrt(Pi)*3^(n-1/2)*n^(3/2)). (End)
G.f.: A(x) satisfies A(x)=x*(1+A(x))/((1-A(x))*(1+A(x)^3)). - Vladimir Kruchinin, Jun 01 2014
G.f. is series reversion of (x - x^2) * (1 - x + x^2) = x - 2*x^2 + 2*x^3 - x^4. - Michael Somos, Jun 01 2014
From Peter Bala, Aug 22 2024: (Start)
G.f. A(x) = 1 - 1/c(x*c(x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
Sum_{n >= 1} a(n)*y^n = x*c(x), where y = x*(1 - x). (End)

More terms from Robert G. Wilson v, Jun 28 2007

A126694 Expansion of g.f.: 1/(1 - 7xc(x)), where c(x) is the g.f. for A000108.

Original entry on oeis.org

1, 7, 56, 455, 3710, 30282, 247254, 2019087, 16488710, 134656130, 1099686056, 8980749862, 73342721956, 598965319960, 4891549246290, 39947649057855, 326239122661830, 2664286127154330, 21758336553841440, 177693081299126610

Offset: 0

Philippe Deléham, Feb 14 2007

The Hankel transform of this sequence is 7^n = [1, 7, 49, 343, 2401, ...] . The Hankel transform of the aerated sequence with g.f. 1/(1 - 7*x^2*c(x^2)) is also 7^n.

Numbers have the same parity as the Catalan numbers, that is, a(n) is even except for n of the form 2^m - 1. Follows from c(x) = 1/(1 - x*c(x)) == 1/(1 - 7*x*c(x)) (mod 2). - Peter Bala, Jul 24 2016

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

Cf. A000108, A000984, A007854, A076035, A076036, A127628, A115970.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(7*Sqrt(1-4*x) -5) )); // G. C. Greubel, May 05 2019

CoefficientList[Series[2/(-5+7*Sqrt[1-4*x]), {x, 0, 30}], x] (* G. C. Greubel, May 05 2019 *)

my(x='x+O('x^30)); Vec(2/(7*sqrt(1-4*x) -5)) \\ G. C. Greubel, May 05 2019

(2/(7*sqrt(1-4*x) -5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 05 2019

a(0) = 1, a(n) = (49*a(n-1) - 7*A000108(n-1))/6 for n >= 1.
a(n) = Sum_{k = 0..n} A106566(n,k)*7^k.
a(n) = Sum_{k = 0..n} A039599(n,k)*6^k.
a(n) ~ 5 * 7^(2*n) / 6^(n+1). - Vaclav Kotesovec, Nov 29 2021

a(16) corrected by G. C. Greubel, May 05 2019

A126983 Expansion of 1/(1+x*c(x)), c(x) the g.f. of Catalan numbers A000108.

Original entry on oeis.org

1, -1, 0, -1, -2, -6, -18, -57, -186, -622, -2120, -7338, -25724, -91144, -325878, -1174281, -4260282, -15548694, -57048048, -210295326, -778483932, -2892818244, -10786724388, -40347919626, -151355847012, -569274150156

Offset: 0

Philippe Deléham, Mar 21 2007

sign

Hankel transform is (-1)^n.

Catalan transform of A033999. - R. J. Mathar, Nov 11 2008

Fung Lam, Table of n, a(n) for n = 0..1500
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.

Cf. A000108, A000957, A039599, A064310, A106566.

[1] cat [(-1/2)^n*(1 +(&+[(-2)^k*Binomial(2*k,k)/(k+1): k in [0..n-1]])): n in [1..30]]; // G. C. Greubel, Feb 27 2019

Table[(-1/2)^n*(1 + Sum[ CatalanNumber[k]*(-2)^k, {k, 0, n-1}]), {n, 0, 30}] (* G. C. Greubel, Feb 27 2019 *)

{a(n) = (-1/2)^n*(1+sum(k=0,n-1, (-2)^k*binomial(2*k,k)/(k+1)))};
vector(30, n, n--; a(n)) \\ G. C. Greubel, Feb 27 2019

from itertools import count, islice
def A126983_gen(): # generator of terms
    yield from (1, -1, 0)
    a, c = 0, 1
    for n in count(1):
        yield (a:=-a-(c:=c*((n<<2)+2)//(n+2))>>1)
A126983_list = list(islice(A126983_gen(),20)) # Chai Wah Wu, Apr 27 2023

[1] + [(-1/2)^n*(1 +sum((-2)^k*catalan_number(k) for k in (0..n-1))) for n in (1..30)] # G. C. Greubel, Feb 27 2019

a(n) = (-1)^n*A064310(n).
a(n) = Sum_{k=0..n} A039599(n,k)*(-2)^k.
From Philippe Deléham, Nov 15 2009: (Start)
a(n) = Sum_{k=0..n} A106566(n,k)*(-1)^k, a(0)=1.
a(n) = -A000957(n) for n>0. (End)
Recurrence: 2*(n+2)*a(n+2) = (7*n+2)*a(n+1) + 2*(2*n+1)*a(n). - Fung Lam, May 07 2014
a(n) ~ -2^(2n)/sqrt(Pi*n^3)/9. - Fung Lam, May 07 2014

A144706 Central coefficients of the triangle A132047.

Original entry on oeis.org

1, 6, 18, 60, 210, 756, 2772, 10296, 38610, 145860, 554268, 2116296, 8112468, 31201800, 120349800, 465352560, 1803241170, 7000818660, 27225405900, 106035791400, 413539586460, 1614773623320, 6312296891160, 24700292182800, 96742811049300, 379231819313256

Offset: 0

Paul Barry, Sep 19 2008

Hankel transform is A144708.

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

Cf. A000108, A010686, A039599, A082505, A106566, A132047, A144708.

[n eq 0 select 1 else 3*(n+1)*Catalan(n): n in [0..40]]; // G. C. Greubel, Jun 16 2022

Table[3*Binomial[2n,n] -2*Boole[n==0], {n,0,40}] (* G. C. Greubel, Jun 16 2022 *)

a(n) = if(n, 3*binomial(2*n, n), 1) \\ Charles R Greathouse IV, Oct 23 2023

[3*binomial(2*n, n) -2*bool(n==0) for n in (0..40)] # G. C. Greubel, Jun 16 2022

G.f.: 3/sqrt(1-4*x) - 2;
a(n) = 3*binomial(2*n, n) - 2*0^n.
From Philippe Deléham, Oct 30 2008: (Start)
a(n) = Sum_{k=0..n} A039599(n,k)*A010686(k).
a(n) = Sum_{k=0..n} A106566(n,k)*A082505(k+1). (End)
D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1). - R. J. Mathar, Nov 30 2012
E.g.f.: -2 + 3*exp(2*x)*BesselI(0, 2*x). - G. C. Greubel, Jun 16 2022

A115970 Expansion of 1/(4sqrt(1-4x) - 3).

Original entry on oeis.org

1, 8, 72, 656, 5992, 54768, 500688, 4577568, 41851560, 382641200, 3498428272, 31985610720, 292439802256, 2673735097184, 24445577182368, 223502416896576, 2043450657688872, 18682977401318064, 170815793235313968

Offset: 0

Paul Barry, Feb 03 2006

The g.f. is A(x)^2/(2*A(x)-A(x)^2) where A(x) is the g.f. of A076035.

The Hankel transform of this sequence is 8^n = [1, 8, 64, 512, 4096, ...]; the Hankel transform of the aerated sequence with g.f. 1/(1-8*x^2*c(x^2)) is also 8^n. - Philippe Deléham, Feb 13 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 1/(4*Sqrt(1-4*x)-3) )); // G. C. Greubel, May 05 2019

CoefficientList[Series[1/(4*Sqrt[1-4*x]-3), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

my(x='x+O('x^20)); Vec(1/(4*sqrt(1-4*x)-3)) \\ G. C. Greubel, May 05 2019

(1/(4*sqrt(1-4*x)-3)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 05 2019

G.f.: 1/(1-8*x*c(x)), where c(x) is the g.f. of A000108.
a(n) = Sum_{k=0..n} A106566(n, k)*8^k.
From Philippe Deléham, Feb 13 2007: (Start)
a(n) = (64*a(n-1) - 8*A000108(n-1))/7.
a(n) = Sum_{k=0..n} A039599(n,k)*7^k.
a(n) = Sum_{k=0..n} A106566(n,k)*8^k. (End)
D-finite with recurrence: 7*n*a(n) = 2*(46*n-21)*a(n-1) - 128*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 3*2^(6*n+1)/7^(n+1). - Vaclav Kotesovec, Oct 19 2012

cp's OEIS Frontend

A064310 Generalized Catalan numbers C(-1; n).

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A098474 Triangle read by rows, T(n,k) = C(n,k)*C(2*k,k)/(k+1), n >= 0, 0 <= k <= n.

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A026726 a(n) = T(2n,n), T given by A026725.

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A076036 G.f.: 1/(1 - 5*x*C(x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) = g.f. for the Catalan numbers A000108.

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A109262 A Catalan transform of the Fibonacci numbers.

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A121988 Number of vertices of the n-th multiplihedron.

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A126694 Expansion of g.f.: 1/(1 - 7*x*c(x)), where c(x) is the g.f. for A000108.

A098474 Triangle read by rows, T(n,k) = C(n,k)C(2k,k)/(k+1), n >= 0, 0 <= k <= n.

A076036 G.f.: 1/(1 - 5xC(x)), where C(x) = (1 - sqrt(1 - 4x))/(2x) = g.f. for the Catalan numbers A000108.

A126694 Expansion of g.f.: 1/(1 - 7xc(x)), where c(x) is the g.f. for A000108.

A115970 Expansion of 1/(4sqrt(1-4x) - 3).