A015083 -id:A015083 - cp's OEIS frontend

A377132 The expansion of 1/(1 - G(x)*x), where G(x) is the ordinary generating function of the Carlitz-Riordan q-Catalan numbers for q = 2 (A015083).

1, 1, 2, 6, 28, 228, 3592, 113880, 7267952, 929696784, 237968445472, 121835841547872, 124758916812038592, 255505766282965942848, 1046551115668335283290240, 8573345713494489300568753536, 140465691975467799273799959144192, 4602779760325164559879331800453222656, 301647773810532495378626041621616755442176, 39537576990478498231890121766124629197694682624

Offset: 0

Thomas Scheuerle, Oct 19 2024

nonn

Cf. A002412, A015083, A016061, A376923.

A376923(n, k) = if(n < 0, return(0),return(if(k == 0, return(2^n), T(n-1, k) + 2^(n-1)*T(n-2, k-1) )))
a(n) = if(n==0, 1, sum(k=0, n-1, a(n-k-1)*A376923(n-1, k)*(-1)^k))

O.g.f.: Continued fraction expansion 1/(1 - x/(1 - 2^0*x/(1 - 2^1*x/(1 - 2^2*x/(1 - 2^3*x/(...)))))).
a(n) = Sum_{k=0..n-1} a(n - k - 1)*A376923(n - 1, k)*(-1)^k, with a(0) = 1.
The Hankel determinant of a(0)..a(2*n) is 2^A016061(n). The Hankel determinant of a(1)..a(2*n+1) is 2^A002412(n).

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

Original entry on oeis.org

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)

A015084 Carlitz-Riordan q-Catalan numbers for q=3.

Original entry on oeis.org

1, 1, 4, 43, 1252, 104098, 25511272, 18649337311, 40823535032644, 267924955577741566, 5274102955963545775864, 311441054994969341088610030, 55171471477692117486494217498280

Offset: 0

Olivier Gérard

nonn

Limit_{n->inf} a(n)/3^((n-1)(n-2)/2) = Product{k>=1} 1/(1-1/3^k) = 1.785312341998534190367486296013703535718796... - Paul D. Hanna, Jan 24 2005
It appears that the Hankel transform is 3^A002412(n). - Paul Barry, Aug 01 2008
Hankel transform of the aerated sequence is 3^C(n+1,3). - Paul Barry, Oct 31 2008

			G.f. = 1 + x + 4*x^2 + 43*x^3 + 1252*x^4 + 104098*x^5 + 25511272*x^6 + ...
From _Seiichi Manyama_, Dec 05 2016: (Start)
a(1) = 1,
a(2) = 3^1 + 1 = 4,
a(3) = 3^3 + 3^2 + 2*3^1 + 1 = 43,
a(4) = 3^6 + 3^5 + 2*3^4 + 3*3^3 + 3*3^2 + 3*3^1 + 1 = 1252. (End)

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), A015098 (q=-3), A015097 (q=-2), A090192 (q=-1), A000108 (q=1), A015083 (q=2), this sequence (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 A090182, A290759.

A015084 := proc(n)
    option remember;
    if n = 1 then
        1;
    else
    add(3^(i-1)*procname(i)*procname(n-i),i=1..n-1) ;
    end if;
end proc: # R. J. Mathar, Sep 29 2012

a[n_] := a[n] = Sum[3^i*a[i]*a[n -i -1], {i, 0, n -1}]; a[0] = 1; Array[a, 16, 0] (* Robert G. Wilson v, Dec 24 2016 *)
m = 13; ContinuedFractionK[If[i == 1, 1, -3^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

a(n)=if(n==1,1,sum(i=1,n-1,3^(i-1)*a(i)*a(n-i))) \\ Paul D. Hanna

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 A015084(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). - Paul D. Hanna, Jan 24 2005
a(n) = the upper left term in M^n, M an infinite production matrix as follows:
  1,  3,  0,  0,  0,  0, ...
  1,  3,  9,  0,  0,  0, ...
  1,  3,  9, 27,  0,  0, ...
  1,  3,  9, 27, 81,  0, ...
  ... - Gary W. Adamson, Jul 14 2011
G.f.: T(0), where T(k) = 1 - x*3^k/(x*3^k - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 18 2013

More terms from Paul D. Hanna, Jan 24 2005

Offset changed to 0 by Seiichi Manyama, Dec 05 2016

A015085 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=4.

Original entry on oeis.org

1, 1, 5, 89, 5885, 1518897, 1558435125, 6386478643785, 104648850228298925, 6858476391221411106209, 1797922152786660462507074405, 1885261615172756172119161342909753

Offset: 0

Olivier Gérard

nonn

			G.f. = 1 + x + 5*x^2 + 89*x^3 + 5885*x^4 + 1518897*x^5 + 1558435125*x^6 + ...
From _Seiichi Manyama_, Dec 05 2016: (Start)
a(1) = 1,
a(2) = 4^1 + 1 = 5,
a(3) = 4^3 + 4^2 + 2*4^1 + 1 = 89,
a(4) = 4^6 + 4^5 + 2*4^4 + 3*4^3 + 3*4^2 + 3*4^1 + 1 = 5885. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..58
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), A015097 (q=-2), A090192 (q=-1), A000108 (q=1), A015083 (q=2), A015084 (q=3), this sequence (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 A090182, A290759.

a[n_] := a[n] = Sum[4^i*a[i]*a[n -i -1], {i, 0, n -1}]; a[0] = 1; Array[a, 16, 0] (* Robert G. Wilson v, Dec 24 2016 *)
m = 12; ContinuedFractionK[If[i == 1, 1, -4^(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 A015085(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 26 2016
a(n) ~ c * 2^(n*(n-1)), where c = Product{j>=1} 1/(1-1/4^j) = 1/QPochhammer(1/4) = 1.4523536424495970158347130224852748733612279788... - Vaclav Kotesovec, Nov 03 2021

Offset changed to 0 by Seiichi Manyama, Dec 05 2016

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

A015086 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=5.

Original entry on oeis.org

1, 1, 6, 161, 20466, 12833546, 40130703276, 627122621447281, 48995209411107768186, 19138851672289046707772366, 37380607950584029444762130426196, 365045074278810327614287737714877590426, 17824467247610520516685844671190387550839429556, 4351676609772600016156555731067955626656370700291086836

Offset: 0

Olivier Gérard

nonn

			G.f. = 1 + x + 6*x^2 + 161*x^3 + 20466*x^4 + 12833546*x^5 + 40130703276*x^6 + ...

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), 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), this sequence (q=5), A015089 (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
Column k=5 of A090182, A290759.

a[n_] := a[n] = Sum[5^i*a[i]*a[n -i -1], {i, 0, n -1}];
a[0] = 1; Array[a, 12, 0] (* Robert G. Wilson v, 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 A015086(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 26 2016

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

A015089 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=6.

Original entry on oeis.org

1, 1, 7, 265, 57799, 75025897, 583552122727, 27227375795690569, 7621977131953256556295, 12802009986716861649949951657, 129014790439200398432389878440405671

Offset: 0

Olivier Gérard

nonn

			G.f. = 1 + x + 7*x^2 + 265*x^3 + 57799*x^4 + 75025897*x^5 + 583552122727*x^6 + ...

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), 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), this sequence (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
Column k=6 of A090182, A290759.

a[n_] := a[n] = Sum[6^i*a[i]*a[n -i -1], {i, 0, n -1}]; a[0] = 1; Array[a, 16, 0] (* Robert G. Wilson v, 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 A015089(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 26 2016

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

A015091 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=7.

Original entry on oeis.org

1, 1, 8, 407, 140456, 337520898, 5673390747984, 667480099386451779, 549699898523248769128232, 3168911624115201777713785471406, 127877020635106970108300418456422667248

Offset: 0

Olivier Gérard

nonn

			G.f. = 1 + x + 8*x^2 + 407*x^3 + 140456*x^4 + 337520898*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), 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), this sequence (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
Column k=7 of A090182, A290759.

a[n_] := a[n] = Sum[7^i*a[i]*a[n -i -1], {i, 0, n -1}]; a[0] = 1; Array[a, 16, 0] (* Robert G. Wilson v, Dec 24 2016 *)
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 A015091(n)
  A(7, n)
end # Seiichi Manyama, Dec 24 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 26 2016

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

cp's OEIS Frontend

A377132 The expansion of 1/(1 - G(x)*x), where G(x) is the ordinary generating function of the Carlitz-Riordan q-Catalan numbers for q = 2 (A015083).

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

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A015084 Carlitz-Riordan q-Catalan numbers for q=3.

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A015085 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=4.

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

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

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

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