A290759 -id:A290759 - cp's OEIS frontend

A015083 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=2.

1, 1, 3, 17, 171, 3113, 106419, 7035649, 915028347, 236101213721, 121358941877763, 124515003203007345, 255256125633703622475, 1046039978882750301409545, 8571252355254982356001107795, 140448544236464264647066322058465, 4602498820363674769217316088142020635

Offset: 0

Olivier Gérard

nonn

Limit_{n->inf} a(n)/2^((n-1)(n-2)/2) = Product{k>=1} 1/(1-1/2^k) = 3.462746619455... (cf. A065446). - Paul D. Hanna, Jan 24 2005
It appears that the Hankel transform is 2^A002412(n). - Paul Barry, Aug 01 2008
Hankel transform of aerated sequence is A125791. - Paul Barry, Dec 15 2010

			G.f. = 1 + x + 3*x^2 + 17*x^3 + 171*x^4 + 3113*x^5 + 106419*x^6 + 7035649*x^7 + ...
From _Seiichi Manyama_, Dec 05 2016: (Start)
a(1) = 1,
a(2) = 2^1 + 1 = 3,
a(3) = 2^3 + 2^2 + 2*2^1 + 1 = 17,
a(4) = 2^6 + 2^5 + 2*2^4 + 3*2^3 + 3*2^2 + 3*2^1 + 1 = 171. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..81
J. Fürlinger, J. Hofbauer, q-Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 40, Issue 2, November 1985, Pages 248-264.
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Cf. A065446, 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), this sequence (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 A090182, A290759.

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

a(n)=if(n==0,1,sum(i=0,n-1,2^i*a(i)*a(n-1-i))) \\  Paul D. Hanna

{a(n) = my(A); if( n<1, n==0, A = vector(n, i, 1); for(k=0, n-1, A[k+1] = if( k<1, 1, A[k]*(1+2^k) + sum(i=1, k-1, 2^i * A[i] * A[k-i]))); A[n])}; /* Michael Somos, Jan 30 2005 */

{a(n) = my(A); if( n<0, 0, A = O(x); for(k=1, n, A = 1 / (1 - x * subst(A, x, 2*x))); polcoeff(A, n))}; /* Michael Somos, Jan 30 2005 */

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 A015083(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. satisfies: A(x) = 1 / (1 - x*A(2*x)) = 1/(1-x/(1-2*x/(1-2^2*x/(1-2^3*x/(1-...))))) (continued fraction). - Paul D. Hanna, Jan 24 2005
G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=0..n-1} 2^k*x*A(2^k*x). - Paul D. Hanna, May 17 2010
a(n) = the upper left term in M^(n-1), M = the infinite square production matrix:
  1,  2,  0,  0,  0, ...
  1,  2,  4,  0,  0, ...
  1,  2,  4,  8,  0, ...
  1,  2,  4,  8, 16, ...
  ...
Also, a(n+1) = sum of top row terms of M^(n-1). Example: top row of M^3 = (17, 34, 56, 64, 0, 0, 0, ...); where a(4) = 17 and a(5) = 171 = (17 + 34 + 56 + 64). - Gary W. Adamson, Jul 14 2011
G.f.: T(0), where T(k) = 1 - x*(2^k)/(x*(2^k) - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2013

Offset changed to 0 by Seiichi Manyama, Dec 05 2016

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

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

A015092 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=8.

Original entry on oeis.org

1, 1, 9, 593, 304857, 1249312673, 40939981188777, 10732252327798007281, 22507185898866512901924729, 377607964391970470904956530918721, 50681683810611444451901001718927186370889

Offset: 0

Olivier Gérard

nonn

			G.f. = 1 + x + 9*x^2 + 593*x^3 + 304857*x^4 + 1249312673*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..47
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), 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), this sequence (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
Column k=8 of A090182, A290759.

a[n_] := a[n] = Sum[8^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, -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 A015092(n)
  A(8, n)
end # Seiichi Manyama, Dec 24 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 26 2016

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

A015093 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=9.

Original entry on oeis.org

1, 1, 10, 829, 606070, 3977651242, 234884294434900, 124827614155955343925, 597046858511123656669455550, 25700910736350654917922270058287454, 9957059456624152426469878400757673046606860

Offset: 0

Olivier Gérard

nonn

			G.f. = 1 + x + 10*x^2 + 829*x^3 + 606070*x^4 + 3977651242*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), 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), this sequence (q=9), A015095 (q=10), A015096 (q=11).
Column k=9 of A090182, A290759.

a[n_] := a[n] = Sum[9^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, -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 A015093(n)
  A(9, n)
end # Seiichi Manyama, Dec 24 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 27 2016

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

A015095 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=10.

Original entry on oeis.org

1, 1, 11, 1121, 1123331, 11235577641, 1123580257785051, 1123582505161487376561, 11235827298801257861061293171, 1123582752351801734250808539216885881

Offset: 0

Olivier Gérard

nonn

			G.f. = 1 + x + 11*x^2 + 1121*x^3 + 1123331*x^4 + 11235577641*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), 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), A015091 (q=7), A015092 (q=8), A015093 (q=9), this sequence (q=10), A015096 (q=11).
Column k=10 of A090182, A290759.

a[n_] := a[n] = Sum[10^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 = 10; 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 A015095(n)
  A(10, n)
end # Seiichi Manyama, Dec 24 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 27 2016

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

A015096 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=11.

Original entry on oeis.org

1, 1, 12, 1475, 1966284, 28792327202, 4637090716230072, 8214898341126993790759, 160085145151052208703206236460, 34315672899472590258644379240786601502

Offset: 0

Olivier Gérard

nonn

			G.f. = 1 + x + 12*x^2 + 1475*x^3 + 1966284*x^4 + 28792327202*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..44
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Masterarbeit, University of Vienna. Fakultät für Mathematik, 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), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), this sequence (q=11).
Column k=11 of A090182, A290759.

a[n_] := a[n] = Sum[11^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 *)

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 A015096(n)
  A(11, n)
end # Seiichi Manyama, Dec 24 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=11 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(11*x)) = 1/(1-x/(1-11*x/(1-11^2*x/(1-11^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 27 2016

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

cp's OEIS Frontend

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