A290789 -id:A290789 - cp's OEIS frontend

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

Philippe Deléham, Jan 22 2004

sign

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

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

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

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.

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

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 *)

{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 */

Vec((2*x - 1 + sqrt(1+4*x^2))/(2*x) + O(x^50)) \\ G. C. Greubel, Dec 24 2016

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

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

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)

A015097 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-2.

Original entry on oeis.org

1, 1, -1, -7, 47, 873, -26433, -1749159, 220526159, 56904690761, -29022490524961, -29777360924913095, 60924625361199230575, 249669263740090899509545, -2044791574538659983034398465, -33505955988983997787211823466215

Offset: 0

Olivier Gérard

sign

			G.f. = 1 + x - x^2 - 7*x^3 + 47*x^4 + 873*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..81
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

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), this sequence (q=-2), A090192 (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=2 of A290789.

m = 16;
ContinuedFractionK[If[i == 1, 1, (-1)^(i+1) 2^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

l=[1]
for n in range(1, 21):
    l.append(sum([(-2)**i*l[i]*l[n - 1 - i] for i in range(n)]))
print(l) # Indranil Ghosh, Aug 14 2017

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 A015097(n)
  A(-2, n)
end # Seiichi Manyama, Dec 24 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-2 and a(0)=1.
G.f: 1/(1-x/(1+2x/(1-4x/(1+8x/(1-16x/(1+... (continued fraction). - Paul Barry, Jan 15 2009
G.f. satisfies: A(x) = 1 / (1 - x*A(-2*x)). - Seiichi Manyama, Dec 27 2016

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

A015098 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-3.

Original entry on oeis.org

1, 1, -2, -23, 586, 48778, -11759396, -8596478231, 18783386191762, 123275424165263086, -2426183754235085042972, -143268577734839493464012630, 25379312219817753259837452498340

Offset: 0

Olivier Gérard

sign

			G.f. = 1 + x - 2*x^2 - 23*x^3 + 586*x^4 + 48778*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..65
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

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), this sequence (q=-3), A015097 (q=-2), A090192 (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=3 of A290789.

a[1] := 1; a[n_] := a[n] = Sum[(-3)^(i - 1)*a[i]*a[n - i], {i, 1, n - 1}]; Array[a, 20, 1] (* G. C. Greubel, Dec 24 2016 *)
m = 13; ContinuedFractionK[If[i == 1, 1, (-1)^(i+1) 3^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

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 A015098(n)
  A(-3, n)
end # Seiichi Manyama, Dec 24 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-3 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(-3*x)) = 1/(1-x/(1+3*x/(1-3^2*x/(1+3^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 27 2016

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

A015099 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-4.

Original entry on oeis.org

1, 1, -3, -55, 3429, 885137, -904638963, -3707218743911, 60731665539301365, 3980231929565571675617, -1043385959026442521712292579, -1094071562179856506263860787078039

Offset: 0

Olivier Gérard

sign

			G.f. = 1 + x - 3*x^2 - 55*x^3 + 3429*x^4 + 885137*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..58
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers

Cf. A227543.
Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), this sequence (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (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=4 of A290789.

a[1] := 1; a[n_] := a[n] = Sum[(-4)^(i - 1)*a[i]*a[n - i], {i, 1, n - 1}]; Array[a, 20, 1] (* G. C. Greubel, Dec 24 2016 *)

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 A015099(n)
  A(-4, n)
end # Seiichi Manyama, Dec 24 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-4 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(-4*x)) = 1/(1-x/(1+4*x/(1-4^2*x/(1+4^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 27 2016

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

A015100 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-5.

Original entry on oeis.org

1, 1, -4, -109, 13436, 8425506, -26312994024, -411193252871529, 32123650782112689116, 12548365338592689141400286, -24508500955561451477156078353144

Offset: 0

Olivier Gérard

sign

			G.f. = 1 + x - 5*x^2 - 191*x^3 + 40915*x^4 + 53110057*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..53
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Cf. A227543.
Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), this sequence (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (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=5 of A290789.

a[1] := 1; a[n_] := a[n] = Sum[(-5)^(i - 1)*a[i]*a[n - i], {i, 1, n - 1}]; Array[a, 20, 1] (* G. C. Greubel, Dec 24 2016 *)
m = 11; ContinuedFractionK[If[i == 1, 1, -(-5)^(i - 2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

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 A015100(n)
  A(-5, n)
end # Seiichi Manyama, Dec 24 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-5 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(-5*x)) = 1/(1-x/(1+5*x/(1-5^2*x/(1+5^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 27 2016

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

A015102 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-6.

Original entry on oeis.org

1, 1, -5, -191, 40915, 53110057, -412878084725, -19264066381851695, 5392667163887921078275, 9057620836725683164283293369, -91279931160615494871228103624209605

Offset: 0

Olivier Gérard

sign

			G.f. = 1 + x - 5*x^2 - 191*x^3 + 40915*x^4 + 53110057*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..51
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Cf. A227543.
Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), this sequence (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (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=6 of A290789.

a[1] := 1; a[n_] := a[n] = Sum[(-6)^(i - 1)*a[i]*a[n - i], {i, 1, n - 1}]; Array[a, 12, 1] (* G. C. Greubel, Dec 24 2016 *)
m = 11; ContinuedFractionK[If[i == 1, 1, -(-6)^(i - 2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

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 A015102(n)
  A(-6, n)
end # Seiichi Manyama, Dec 24 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-6 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(-6*x)) = 1/(1-x/(1+6*x/(1-6^2*x/(1+6^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 27 2016

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

A015103 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-7.

Original entry on oeis.org

1, 1, -6, -307, 104742, 251699498, -4229811552588, -497641562809372379, 409828230340907182689774, 2362579011761419853955236859806, -95338580221916838164306067991935130836

Offset: 0

Olivier Gérard

sign

			G.f. = 1 + x - 6*x^2 - 307*x^3 + 104742*x^4 + 251699498*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..49
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Cf. A227543.
Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), this sequence (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (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=7 of A290789.

m = 11; ContinuedFractionK[If[i == 1, 1, -(-7)^(i - 2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

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 A015103(n)
  A(-7, n)
end # Seiichi Manyama, Dec 25 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-7, and a(0) = 1.

G.f. satisfies: A(x) = 1 / (1 - x*A(-7*x)) = 1/(1-x/(1+7*x/(1-7^2*x/(1+7^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 28 2016

Offset changed to 0 by Seiichi Manyama, Dec 25 2016

A015105 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-8.

Original entry on oeis.org

1, 1, -7, -463, 236201, 967959393, -31716161292711, -8314264834902720111, 17436260499054618815283977, 292531943497569504642889779448001, -39262972238604561849241166988994502971207, -42158295505666574904992534735221817668906507523855

Offset: 0

Olivier Gérard

sign

			G.f. = 1 + x - 7*x^2 - 463*x^3 + 236201*x^4 + 967959393*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..47
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Cf. A227543.
Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), this sequence (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (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=8 of A290789.

m = 11; ContinuedFractionK[If[i == 1, 1, -(-8)^(i - 2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

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 A015105(n)
  A(-8, n)
end # Seiichi Manyama, Dec 25 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-8 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(-8*x)) = 1/(1-x/(1+8*x/(1-8^2*x/(1+8^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 28 2016

Offset changed to 0 by Seiichi Manyama, Dec 25 2016

A015106 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-9.

Original entry on oeis.org

1, 1, -8, -665, 483544, 3173511682, -187386353065808, -99585165693268026701, 476312561203989614441440600, 20503694883570579788445502041773422, -7943551457092331370323478258038812629918704

Offset: 0

Olivier Gérard

sign

			G.f. = 1 + x - 8*x^2 - 665*x^3 + 483544*x^4 + 3173511682*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..46
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Cf. A227543.
Cf. A015108 (q=-11), A015107 (q=-10), this sequence (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (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=9 of A290789.

m = 11; ContinuedFractionK[If[i == 1, 1, -(-9)^(i - 2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

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 A015106(n)
  A(-9, n)
end # Seiichi Manyama, Dec 25 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-9 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(-9*x)) = 1/(1-x/(1+9*x/(1-9^2*x/(1+9^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 28 2016

Offset changed to 0 by Seiichi Manyama, Dec 25 2016

A015107 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-10.

Original entry on oeis.org

1, 1, -9, -919, 917271, 9174563561, -917438025443049, -917439860513400673559, 9174396770273536422744011031, 917439695376166450708460281823359721, -917439693541287252616828116888122637934368489

Offset: 0

Olivier Gérard

sign

			G.f. = 1 + x - 9*x^2 - 919*x^3 + 917271*x^4 + 9174563561*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..45
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Cf. A227543.
Cf. A015108 (q=-11), this sequence (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), A090192 (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=10 of A290789.

m = 11; ContinuedFractionK[If[i == 1, 1, -(-10)^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

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 A015107(n)
  A(-10, n)
end # Seiichi Manyama, Dec 25 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-10 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(-10*x)) = 1/(1-x/(1+10*x/(1-10^2*x/(1+10^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 28 2016

Offset changed to 0 by Seiichi Manyama, Dec 25 2016

cp's OEIS Frontend

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