A064334 -id:A064334 - cp's OEIS frontend

A064338 Row sums of signed triangle A064334.

1, 2, 3, 3, 3, 4, -6, 49, -178, 76, 8633, -124131, 1202662, -8252662, 16009623, 730544913, -17236420029, 256679586178, -2787185606474, 15947981601793, 215110909342463, -9723413157539188, 216257810122284716, -3515999949642686709

Offset: 0

Wolfdieter Lang, Sep 21 2001

sign

Robert Israel, Table of n, a(n) for n = 0..488

Cf. A064339 (row sums of unsigned triangle A064334).

f:= proc(n) local k; 1 + add(simplify(hypergeom([1-n+k,n-k],[-n+k],-k)),k=0..n-1) end proc:
map(f, [$0..50]); # Robert Israel, Jan 03 2019

a[n_] := If[n == 0, 1, Sum[(n-k-j)*Binomial[n-k-1+j, j]* (-k)^j/(n-k), {k, 1, n-1}, {j, 0, n-k-1}] + 2];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 06 2023 *)

a(n) = 1 + Sum_{0<=k<=n-1} hypergeom([1-n+k,n-k],[-n+k],-k). - Robert Israel, Jan 03 2019

A064339 Row sums of unsigned triangle A064334.

Original entry on oeis.org

1, 2, 3, 3, 5, 12, 50, 289, 2032, 16384, 147817, 1471633, 16002504, 188441358, 2385787007, 32281229769, 464479506181, 7076959159858, 113762276632726, 1923164965554837, 34092037061635649, 632112131123669460

Offset: 0

Wolfdieter Lang, Sep 21 2001

nonn

A064338 (row sums of signed triangle A064334).

A064062 Generalized Catalan numbers C(2; n).

Original entry on oeis.org

1, 1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 210731011, 1493303293, 10678370307, 76957679613, 558403682307, 4075996839933, 29909606989827, 220510631755773, 1632599134961667, 12133359132082173

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=2, beta=1 (or alpha=1, beta=2).
a(n) = number of Dyck n-paths (A000108) in which each upstep (U) not at ground level is colored red (R) or blue (B). For example, a(3)=3 counts URDD, UBDD, UDUD (D=downstep). - David Callan, Mar 30 2007
The Hankel transform of this sequence is A002416. - Philippe Deléham, Nov 19 2007
The sequence a(n)/2^n, with g.f. 1/(1-xc(x)/2), has Hankel transform 1/2^n. - Paul Barry, Apr 14 2008
The REVERT transform of the odd numbers [1,3,5,7,9,...] is [1, -3, 13, -67, 381, -2307, 14589, -95235, 636925, ...] - N. J. A. Sloane, May 26 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Abate and W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, example section 3.
Jean-Luc Baril, Sergey Kirgizov, and Mehdi Naima, A lattice on Dyck paths close to the Tamari lattice, arXiv:2309.00426 [math.CO], 2023.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From _N. J. A. Sloane_, Oct 08 2012
J. Bloom and S. Elizalde, Pattern avoidance in matchings and partitions, arXiv:1211.3442 [math.CO] (2012) Theorem 6.1.
N. Bonichon, C. Gavoille and N. Hanusse, Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation, In Proceedings of WG'03, volume 2880 of LNCS, pp. 81-92, 2003.
Alexander Burstein, Sergi Elizalde and Toufik Mansour, Restricted Dumont permutations, Dyck paths and noncrossing partitions, arXiv:math/0610234 [math.CO], 2006.
Xiang-Ke Chang, X.-B. Hu, H. Lei and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
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).
Simone Fioravanti, Steve Hanneke, Shay Moran, Hilla Schefler, and Iska Tsubari, Ramsey Theorems for Trees and a General 'Private Learning Implies Online Learning' Theorem, arXiv:2407.07765 [cs.LG], 2024. See p. 44.
Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics 24(2) (2017), #P2.3.
N. J. A. Sloane, Transforms
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011-2012.

Generalized Catalan numbers C(m; n): A000012 (m = 0), A000108 (m = 1), A064063 (m = 3) and A064087 - A064093 (m = 4 thru 10); A064310 (m = -1), A064311 (m = -2) and A064325 - A064333 (m = -3 thru -11).

Cf. A064334, A119529.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (3 - Sqrt(1-8*x))/(2*(1+x)) )); // G. C. Greubel, Sep 27 2024

1, seq(simplify(hypergeom([1-n,n],[-n],2)), n=1..100); # Robert Israel, Nov 30 2014

a[0]=1; a[1]=1; a[n_]/;n>=2 := a[n] = a[n-1] + Sum[(a[k] + a[k-1])a[n-k],{k,n-1}]; Table[a[n],{n,0,10}] (* David Callan, Aug 27 2009 *)
a[n_] := 2*Sum[ (-1)^j*2^(n-j-1)*Binomial[2*(n-j-1), n-j-1]/(n-j), {j, 0, n-1}] + (-1)^n; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 03 2013 *)

{a(n)=polcoeff((3-sqrt(1-8*x+x*O(x^n)))/(2+2*x),n)}

{a(n)=local(A=1+x); for(i=1, n, A=1+A^4*intformal(1/(A^2+x*O(x^n)))); polcoeff(A, n)} \\ Paul D. Hanna, Dec 24 2013
for(n=0, 25, print1(a(n), ", "))

{a(n)=polcoeff(1/(1 - serreverse(x-2*x^2 +x^2*O(x^n))),n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Nov 30 2014

def a(n):
    if n==0: return 1
    return hypergeometric([1-n, n], [-n], 2).simplify()
[a(n) for n in range(22)] # Peter Luschny, Dec 01 2014

G.f.: (1 + 2*x*C(2*x)) / (1+x) = 1/(1 - x*C(2*x)) with C(x) g.f. of Catalan numbers A000108.
a(n) = A062992(n-1) = Sum_{m = 0..n-1} (n-m)*binomial(n-1+m, m)*(2^m)/n, n >= 1, a(0) = 1.
a(n) = Sum_{k = 0..n} A059365(n, k)*2^(n-k). - Philippe Deléham, Jan 19 2004
G.f.: 1/(1-x/(1-2x/(1-2x/(1-2x/(1-.... = 1/(1-x-2x^2/(1-4x-4x^2/(1-4x-4x^2/(1-.... (continued fractions). - Paul Barry, Jan 30 2009
a(n) = (32/Pi)*Integral_{x = 0..1} (8*x)^(n-1)*sqrt(x*(1-x)) / (8*x+1). - Groux Roland, Dec 12 2010
a(n+2) = 8^(n+2)*( c(n+2)-c(1)*c(n+1) - Sum_{i=0..n-1} 8^(-i-2)*c(n-i)*a(i+2) ) with c(n) = Catalan(n+2)/2^(2*n+1). - Groux Roland, Dec 12 2010
a(n) = the upper left term in M^n, M = the production matrix:
  1, 1
  2, 2, 1
  4, 4, 2, 1
  8, 8, 4, 2, 1
  ... - Gary W. Adamson, Jul 08 2011
D-finite with recurrence: n*a(n) + (12-7n)*a(n-1) + 4*(3-2n)*a(n-2) = 0. - R. J. Mathar, Nov 16 2011 (This follows easily from the generating function. - Robert Israel, Nov 30 2014)
G.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^2 dx. - Paul D. Hanna, Dec 24 2013
G.f. satisfies: Integral 1/A(x)^2 dx = x - x^2*G(x), where G(x) is the o.g.f. of A000257, the number of rooted bicubic maps. - Paul D. Hanna, Dec 24 2013
G.f. A(x) satisfies: A(x - 2*x^2) = 1/(1-x). - Paul D. Hanna, Nov 30 2014
a(n) = hypergeometric([1-n, n], [-n], 2) for n > 0. - Peter Luschny, Nov 30 2014
G.f.: (3 - sqrt(1-8*x))/(2*(x+1)). - Robert Israel, Nov 30 2014
a(n) ~ 2^(3*n+1) / (9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 22 2014
O.g.f. A(x) =  1 + series reversion of (x*(1 - x)/(1 + x)^2). Logarithmically differentiating (A(x) - 1)/x gives 3 + 17*x + 111*x^2 + ..., essentially a g.f for A119259. - Peter Bala, Oct 01 2015
From Peter Bala, Jan 06 2022: (Start)
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + ... is a g.f. for A022558.
The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)

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

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

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

A064325 Generalized Catalan numbers C(-3; n).

Original entry on oeis.org

1, 1, -2, 13, -98, 826, -7448, 70309, -686090, 6865150, -70057772, 726325810, -7628741204, 81002393668, -868066319108, 9376806129493, -101988620430938, 1116026661667318, -12277755319108748, 135715825209716038, -1506587474535945788, 16789107646422189868, -187747069029477151328

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.

G. C. Greubel, Table of n, a(n) for n = 0..850

Cf. A064334, A000108.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (7 +Sqrt(1+12*x))/(2*(4-x)) )); // G. C. Greubel, May 03 2019

a[0] = 1;
a[n_] := Sum[(n-m) Binomial[n+m-1, m] (-3)^m/n, {m, 0, n-1}];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jul 30 2018 *)
CoefficientList[Series[(7 +Sqrt[1+12*x])/(2*(4-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

a(n) = if (n==0, 1, sum(m=0, n-1, (n-m)*binomial(n-1+m, m)*(-3)^m/n)); \\ Michel Marcus, Jul 30 2018

my(x='x+O('x^30)); Vec((7 +sqrt(1+12*x))/(2*(4-x))) \\ G. C. Greubel, May 03 2019

def a(n):
    if n == 0: return 1
    return hypergeometric([1-n, n], [-n], -3).simplify()
[a(n) for n in range(24)] # Peter Luschny, Nov 30 2014

((7 +sqrt(1+12*x))/(2*(4-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-3)^m/n.
a(n) = (1/4)^n*(1 + 3*Sum_{k=0..n-1} C(k)*(-3*4)^k), n >= 1, a(0) = 1; with C(n) = A000108(n) (Catalan).
G.f.: (1+3*x*c(-3*x)/4)/(1-x/4) = 1/(1-x*c(-3*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = hypergeometric([1-n, n], [-n], -3) for n>0. - Peter Luschny, Nov 30 2014

A064333 Generalized Catalan numbers C(-11; n).

Original entry on oeis.org

1, 1, -10, 221, -6082, 187386, -6184848, 213843477, -7645509706, 280351640702, -10485617230780, 398467433529298, -15341431926699284, 597149747213056324, -23459916801814723548, 929028306450848244741, -37045540042729366580442

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.

G. C. Greubel, Table of n, a(n) for n = 0..600

Cf. A000108, A064334.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (23 +Sqrt(1+44*x))/(2*(12-x)) )); // G. C. Greubel, May 03 2019

CoefficientList[Series[(23 +Sqrt[1+44*x])/(2*(12-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

my(x='x+O('x^30)); Vec((23 +sqrt(1+44*x))/(2*(12-x))) \\ G. C. Greubel, May 03 2019

((23 +sqrt(1+44*x))/(2*(12-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

a(n) = Sum_{m-0..n-1} (n-m)*binomial(n-1+m, m)*(-11)^m/n.
a(n) = (1/12)^n*(1 + 11*Sum_{k=0..n-1} C(k)*(-11*12)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
G.f.: (1+11*x*c(-11*x)/12)/(1-x/12) = 1/(1-x*c(-11*x)) with c(x) g.f. of Catalan numbers A000108.

A064311 Generalized Catalan numbers C(-2; n).

Original entry on oeis.org

1, 1, -1, 5, -25, 141, -849, 5349, -34825, 232445, -1582081, 10938709, -76616249, 542472685, -3876400305, 27919883205, -202480492905, 1477306676445, -10836099051105, 79861379898165, -591082795606425

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.

Generalized Catalan numbers C(m; n): A000012 (m = 0), A000108 (m = 1), A064062 (m = 2), A064063 (m = 3), A064087 - A064093 (m = 4 thru 10); A064310 (m = -1) and A064325 - A064333 (m = -3 thru -11).

a[n_] := If[n==0, 1, Sum[(n-m)*Binomial[n+m-1, m]*(-2)^m/n, {m,0,n-1}]];
Table[a[n], {n,0,20}] (* Jean-François Alcover, Jun 03 2019 *)

import mpmath
mp.dps = 25; mp.pretty = True
a = lambda n: mpmath.hyp2f1(1-n, n, -n, -2) if n>0 else 1
[int(a(n)) for n in range(21)] # Peter Luschny, Nov 30 2014

a(n) = (1/n) * Sum_{m = 0..n-1} (n-m)*binomial(n-1+m, m)*(-2)^m = ((1/3)^n)*(1 + 2*Sum_{k = 0..n-1} C(k)*(-2*3)^k), for n >= 1, with a(0) := 1, and where C(n) = A000108(n), the Catalan numbers.
G.f.: (1+2*x*c(-2*x)/3)/(1-x/3) = 1/(1-x*c(-2*x)) with c(x) the g.f. of the Catalan numbers A000108.
a(n) = hypergeom([1-n, n], [-n], -2) for n>0. - Peter Luschny, Nov 30 2014
a(n) ~ -(-1)^n * 2^(3*n+1) / (25 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 03 2019
G.f. A(x) = 1 + series_reversion(x*(1 - (m-1)*x)/(1 + x)^2) at m = -2. - Peter Bala, Sep 08 2024

A064327 Generalized Catalan numbers C(-5; n).

Original entry on oeis.org

1, 1, -4, 41, -514, 7206, -108174, 1700721, -27646234, 460887086, -7836596944, 135380098426, -2369445113804, 41925242220616, -748729419265314, 13478117036893281, -244306305241572474, 4455242518055441046, -81683397232911983784, 1504758636166747742286

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.

G. C. Greubel, Table of n, a(n) for n = 0..765

Cf. A064334.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (11 +Sqrt(1+20*x))/(2*(6-x)) )); // G. C. Greubel, May 03 2019

CoefficientList[Series[(11 +Sqrt[1+20*x])/(2*(6-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

my(x='x+O('x^30)); Vec((11 +sqrt(1+20*x))/(2*(6-x))) \\ G. C. Greubel, May 03 2019

def a(n):
    if n==0: return 1
    return hypergeometric([1-n, n], [-n], -5).simplify()
[a(n) for n in range(24)] # Peter Luschny, Nov 30 2014

((11 +sqrt(1+20*x))/(2*(6-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-5)^m/n.
a(n) = (1/6)^n*(1 + 5*Sum_{k=0..n-1} C(k)*(-5*6)^k), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
G.f.: (1+5*x*c(-5*x)/6)/(1-x/6) = 1/(1-x*c(-5*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = hypergeometric([1-n, n], [-n], -5) for n > 0. - Peter Luschny, Nov 30 2014

A064329 Generalized Catalan numbers C(-7; n).

Original entry on oeis.org

1, 1, -6, 85, -1490, 29226, -614004, 13511709, -307448490, 7174776190, -170777485556, 4130050311234, -101192982385844, 2506610481299380, -62668163792277840, 1579300030107459885, -40076101342241993370

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.

G. C. Greubel, Table of n, a(n) for n = 0..690

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (15 +Sqrt(1+28*x))/(2*(8-x)) )); // G. C. Greubel, May 03 2019

CoefficientList[Series[(15 +Sqrt[1+28*x])/(2*(8-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

my(x='x+O('x^30)); Vec((15 +sqrt(1+28*x))/(2*(8-x))) \\ G. C. Greubel, May 03 2019

((15 +sqrt(1+28*x))/(2*(8-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-7)^m/n.
a(n) = (1/8)^n*(1 + 7*Sum_{k=0..n-1} C(k)*(-7*8)^k), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
G.f.: (1+7*x*c(-7*x)/8)/(1-x/8) = 1/(1-x*c(-7*x)) with c(x) g.f. of Catalan numbers A000108.

A064331 Generalized Catalan numbers C(-9; n).

Original entry on oeis.org

1, 1, -8, 145, -3266, 82342, -2223818, 62912809, -1840413050, 55217088622, -1689752866904, 52538652432586, -1655036407913948, 52708355827445800, -1694246075896308110, 54894923324331676345, -1790984858946499478330

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.

G. C. Greubel, Table of n, a(n) for n = 0..640

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (19 +Sqrt(1+36*x))/(2*(10-x)) )); // G. C. Greubel, May 03 2019

CoefficientList[Series[(19 +Sqrt[1+36*x])/(2*(10-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

my(x='x+O('x^30)); Vec((19 +sqrt(1+36*x))/(2*(10-x))) \\ G. C. Greubel, May 03 2019

((19 +sqrt(1+36*x))/(2*(10-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-9)^m/n.
a(n) = (1/10)^n*(1 + 9*Sum_{k=0..n-1} C(k)*(-9*10)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
G.f.: (1+9*x*c(-9*x)/10)/(1-x/10) = 1/(1-x*c(-9*x)) with c(x) g.f. of Catalan numbers A000108.

cp's OEIS Frontend

A064338 Row sums of signed triangle A064334.

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A064339 Row sums of unsigned triangle A064334.

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A064062 Generalized Catalan numbers C(2; n).

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PARI

PARI

PARI

Sage

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

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A064325 Generalized Catalan numbers C(-3; n).

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PARI

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Sage

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A064333 Generalized Catalan numbers C(-11; n).

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Mathematica

PARI

Sage

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A064311 Generalized Catalan numbers C(-2; n).

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