A227543 -id:A227543 - cp's OEIS frontend

A227532 Logarithmic derivative, wrt x, of triangle A227543, as read by terms k=0..n*(n-1)/2 in rows n>=1.

1, 1, 2, 1, 3, 3, 3, 1, 4, 6, 8, 8, 4, 4, 1, 5, 10, 15, 20, 20, 20, 15, 10, 5, 5, 1, 6, 15, 26, 39, 48, 57, 60, 54, 48, 36, 30, 18, 12, 6, 6, 1, 7, 21, 42, 70, 98, 126, 154, 168, 175, 168, 154, 133, 112, 84, 70, 49, 35, 21, 14, 7, 7, 1, 8, 28, 64, 118, 184, 256, 336, 408, 472, 516, 536, 532, 504, 464, 408, 360, 296, 248, 192, 152, 112, 88, 56, 40, 24, 16, 8, 8

Offset: 1

Paul D. Hanna, Jul 14 2013

nonn

			L.g.f.: L(x,q) = x*(1) + x^2*(1 + 2*q)/2 + x^3*(1 + 3*q + 3*q^2 + 3*q^3)/3
+ x^4*(1 + 4*q + 6*q^2 + 8*q^3 + 8*q^4 + 4*q^5 + 4*q^6)/4
+ x^5*(1 + 5*q + 10*q^2 + 15*q^3 + 20*q^4 + 20*q^5 + 20*q^6 + 15*q^7 + 10*q^8 + 5*q^9 + 5*q^10)/5
+ x^6*(1 + 6*q + 15*q^2 + 26*q^3 + 39*q^4 + 48*q^5 + 57*q^6 + 60*q^7 + 54*q^8 + 48*q^9 + 36*q^10 + 30*q^11 + 18*q^12 + 12*q^13 + 6*q^14 + 6*q^15)/6 +...
where exponentiation yields the g.f. of triangle A227543:
exp(L(x,q)) = 1 + x*(1) + x^2*(1 + q) + x^3*(1 + 2*q + q^2 + q^3)
+ x^4*(1 + 3*q + 3*q^2 + 3*q^3 + 2*q^4 + q^5 + q^6)
+ x^5*(1 + 4*q + 6*q^2 + 7*q^3 + 7*q^4 + 5*q^5 + 5*q^6 + 3*q^7 + 2*q^8 + q^9 + q^10)
+ x^6*(1 + 5*q + 10*q^2 + 14*q^3 + 17*q^4 + 16*q^5 + 16*q^6 + 14*q^7 + 11*q^8 + 9*q^9 + 7*q^10 + 5*q^11 + 3*q^12 + 2*q^13 + q^14 + q^15) +...
This triangle begins:
1;
1, 2;
1, 3, 3, 3;
1, 4, 6, 8, 8, 4, 4;
1, 5, 10, 15, 20, 20, 20, 15, 10, 5, 5;
1, 6, 15, 26, 39, 48, 57, 60, 54, 48, 36, 30, 18, 12, 6, 6;
1, 7, 21, 42, 70, 98, 126, 154, 168, 175, 168, 154, 133, 112, 84, 70, 49, 35, 21, 14, 7, 7;
1, 8, 28, 64, 118, 184, 256, 336, 408, 472, 516, 536, 532, 504, 464, 408, 360, 296, 248, 192, 152, 112, 88, 56, 40, 24, 16, 8, 8;
1, 9, 36, 93, 189, 324, 489, 684, 891, 1101, 1305, 1476, 1611, 1683, 1701, 1665, 1593, 1476, 1350, 1197, 1053, 900, 765, 630, 522, 405, 324, 243, 189, 135, 99, 63, 45, 27, 18, 9, 9; ...

Paul D. Hanna, Table of n, a(n) for n = 1..1350 (rows n=1..20 of triangle, flattened).
Stéphane Ouvry and Alexios P. Polychronakos, Exclusion statistics for particles with a discrete spectrum, arXiv:2105.14042 [cond-mat.stat-mech], 2021.

Cf. A227543, A145855, A001700, A153338.

{T(n, k)=local(A=1); for(i=1, n, A=1+x*subst(A, x, q*x)*A +x*O(x^n)); n*polcoeff(polcoeff(log(A), n, x), k, q)}
for(n=1, 10, for(k=0, n*(n-1)/2, print1(T(n, k), ", ")); print(""))

/* By Ramanujan's continued fraction identity: */
{T(n, k)=local(P=1, Q=1);
P=sum(m=0, n+1, q^(m^2)*(-x)^m/prod(k=1, m, 1-q^k) +O(x^(n+2)));
Q=sum(m=0, n+1, q^(m*(m-1))*(-x)^m/prod(k=1, m, 1-q^k) +O(x^(n+2)));
polcoeff(polcoeff(P'/P - Q'/Q, n-1, x), k, q)}
for(n=1, 10, for(k=0, n*(n-1)/2, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Dec 28 2016

L.g.f.: Sum_{k=0..n*(n-1)/2, n>=1} T(n,k)*x^n*q^k/n = Log(G(x,q)) where G(x,q) = 1 + x*G(q*x,q)*G(x,q) is the g.f. of triangle A227543.
Row sums form A001700, the logarithmic derivative of the Catalan numbers.
Sum_{k=0..n*(n-1)/2} T(n,k) = binomial(2*n-1, n-1), for n>=1.
Sum_{k=0..n*(n-1)/2} T(n,k)*(-1)^k = (-1)^[n/2] * binomial(n-1, [(n-1)/2]).
Sum_{k=0..n*(n-1)/2} k*T(n,k) = n*2^(2*n-2) - (2*n-1)*binomial(2*n-2,n-1) = A153338(n), for n>=1.
Sum_{k=0..n*(n-1)/2} T(n,k)*exp(2*Pi*I*k/n) = (-1)^(n-1) for n>=1; i.e., the n-th row sum at q = exp(2Pi*I/n), the n-th root of unity, equals -(-1)^n for n>=1.
Sum_{k=0..[n/2]} T(n, n*k) = A145855(n), the number of n-member subsets of 1..2n-1 whose elements sum to a multiple of n.
L.g.f. satisfies: L'(x,q) = P'(x,q)/P(x,q) - Q'(x,q)/Q(x,q), where
  P(x,q) = Sum_{n>=0} q^(n^2) * (-x)^n / Product_{k=1..n} (1-q^k),
  Q(x,q) = Sum_{n>=0} q^(n*(n-1)) * (-x)^n / Product_{k=1..n} (1-q^k),
due to Ramanujan's continued fraction identity. - Paul D. Hanna, Dec 28 2016

A376527 a(n) = Sum_{k=0..n*(n-1)/2} A227543(n,k)^2 for n >= 0.

Original entry on oeis.org

1, 1, 2, 7, 34, 216, 1610, 13461, 122254, 1183568, 12054498, 127960158, 1405852350, 15901061916, 184381675404, 2184565641269, 26375002217314, 323767457670588, 4033503712929478, 50917059047932592, 650430305337318538, 8398511711996887848, 109507259507469905574, 1440631950110092280386

Offset: 0

Paul D. Hanna, Oct 11 2024

nonn

Compare to binomial(2*n,n)/(n+1) = Sum_{k=0..n*(n-1)/2} A227543(n,k) for n >= 0; that is, the row sums of A227543 equals the Catalan numbers (A000108).
G.f. F(x,q) of triangle A227543 satisfies F(x,q) = 1 + x*F(x,q)*F(q*x,q).
Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0.

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 216*x^5 + 1610*x^6 + 13461*x^7 + 122254*x^8 + 1183568*x^9 + 12054498*x^10 + ...
where coefficient a(n) of x^n in A(x) equals the sum of the square of the terms in row n of triangle A227543, as follows.
a(0) = 1^2 = 1;
a(1) = 1^2 = 1;
a(2) = 1^2 + 1^2 = 2;
a(3) = 1^2 + 2^2 + 1^2 + 1^2 = 7;
a(4) = 1^2 + 3^2 + 3^2 + 3^2 + 2^2 + 1^2 + 1^2 = 34;
a(5) = 1^2 + 4^2 + 6^2 + 7^2 + 7^2 + 5^2 + 5^2 + 3^2 + 2^2 + 1^2 + 1^2 = 216;
a(6) = 1^2 + 5^2 + 10^2 + 14^2 + 17^2 + 16^2 + 16^2 + 14^2 + 11^2 + 9^2 + 7^2 + 5^2 + 3^2 + 2^2 + 1^2 + 1^2 = 1610;
...

Paul D. Hanna, Table of n, a(n) for n = 0..301

Cf. A227543.

\\ From g.f. of A227543, F(x, q) = 1 + x*F(q*x, q)*F(x, q)
{A227543(n, k) = my(F=1); for(i=1, n, F = 1 + x*F*subst(F, x, q*x) +x*O(x^n)); polcoef(polcoef(F, n, x), k, q)}
{a(n) = sum(k=0, n*(n-1)/2, A227543(n, k)^2)}
for(n=0,25, print1(a(n),", "))

\\ faster (using program by Joerg Arndt in A227543)
N=30;
VP=vector(N+1); VP[1] = VP[2] = 1;  \\ one-based; memoization
P(n) = VP[n+1];
for (n=2, N, VP[n+1] = sum( i=0, n-1, P(i) * P(n-1 -i) * x^((i+1)*(n-1-i)) );print1(n,",") );
for(n=0,N, AV=Vec(P(n)); print1(AV*AV~,", "))

a(n) ~ c * 16^n / n^(9/2), where c = 0.430217025951475334005904244213062400539... - Vaclav Kotesovec, Oct 11 2024

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A227532 Logarithmic derivative, wrt x, of triangle A227543, as read by terms k=0..n*(n-1)/2 in rows n>=1.

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A376527 a(n) = Sum_{k=0..n*(n-1)/2} A227543(n,k)^2 for n >= 0.

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

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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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