A015103 -id:A015103 - cp's OEIS frontend

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

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

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

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

A015108 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-11.

Original entry on oeis.org

1, 1, -10, -1231, 1636130, 23957879562, -3858392581773300, -6835385537899011365535, 133202313157282627679850238250, 28553099061411464607955930776882965774

Offset: 0

Olivier Gérard

sign

			G.f. = 1 + x - 10*x^2 - 1231*x^3 + 1636130*x^4 + 23957879562*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..44
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Cf. A227543.
Cf. this sequence (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), A015096 (q=11).
Column k=11 of A290789.

m = 10; ContinuedFractionK[If[i == 1, 1, -(-11)^(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 A015108(n)
  A(-11, n)
end # Seiichi Manyama, Dec 25 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 28 2016

Offset changed to 0 by Seiichi Manyama, Dec 25 2016

cp's OEIS Frontend

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

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

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

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

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

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

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

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