A059365 -id:A059365 - cp's OEIS frontend

A099039 Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.

1, 0, 1, 0, -1, 1, 0, 2, -2, 1, 0, -5, 5, -3, 1, 0, 14, -14, 9, -4, 1, 0, -42, 42, -28, 14, -5, 1, 0, 132, -132, 90, -48, 20, -6, 1, 0, -429, 429, -297, 165, -75, 27, -7, 1, 0, 1430, -1430, 1001, -572, 275, -110, 35, -8, 1, 0, -4862, 4862, -3432, 2002, -1001, 429, -154, 44, -9, 1, 0, 16796, -16796, 11934, -7072, 3640, -1638

Offset: 0

Paul Barry, Sep 23 2004

Row sums are generalized Catalan numbers A064310. Diagonal sums are 0^n+(-1)^n*A030238(n-2). Inverse is A026729, as number triangle. Columns have g.f. (xc(-x))^k=((sqrt(1+4x)-1)/2)^k.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ... ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... ] where DELTA is the operator defined in A084938. - Philippe Deléham, May 31 2005

			Rows begin {1}, {0,1}, {0,-1,1}, {0,2,-2,1}, {0,-5,5,-3,1}, ...
Triangle begins
  1;
  0,    1;
  0,   -1,    1;
  0,    2,   -2,   1;
  0,   -5,    5,  -3,    1;
  0,   14,  -14,   9,   -4,   1;
  0,  -42,   42, -28,   14,  -5,  1;
  0,  132, -132,  90,  -48,  20, -6,  1;
  0, -429,  429, -297, 165, -75, 27, -7, 1;
Production matrix is
  0,  1,
  0, -1,  1,
  0,  1, -1,  1,
  0, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1,
  0, -1,  1, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1, -1,  1,
  0, -1,  1, -1,  1, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1, -1,  1, -1,  1

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021. See (2.10) p. 6.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
D. Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.
Cf. A033184, A000108.
Cf. A106566 (unsigned version), A059365
The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...

T[n_, k_]:= If[n == 0 && k == 0, 1, If[n == 0 && k > 0, 0, (-1)^(n + k)*Binomial[2*n - k - 1, n - k]*k/n]];  Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 31 2017 *)

{T(n,k) = if(n == 0 && k == 0, 1, if(n == 0 && k > 0, 0, (-1)^(n + k)*binomial(2*n - k - 1, n - k)*k/n))};
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 31 2017

T(n, k) = (-1)^(n+k)*binomial(2*n-k-1, n-k)*k/n for 0 <= k <= n with n > 0; T(0, 0) = 1; T(0, k) = 0 if k > 0. - Philippe Deléham, May 31 2005

A064087 Generalized Catalan numbers C(4; n).

Original entry on oeis.org

1, 1, 5, 41, 413, 4641, 55797, 702297, 9137549, 121909457, 1658755685, 22929591433, 321111942781, 4546112358529, 64958195967957, 935566629270201, 13567825195172973, 197957440018622769, 2903721563443327557, 42796201522669935081, 633443408407612143453

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n} in the Derrida et al. 1992 reference (see A064094) for alpha=4, beta=1 (or alpha=1, beta=4).

Cf. A064063 (C(3; n)).

a[0] = 1; a[n_] := Sum[(n - m)*Binomial[n - 1 + m, m]*4^m/n, {m, 0, n - 1}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jul 09 2013 *)

a(n) = if(n<0,0,polcoeff(serreverse((x-3*x^2)/(1+x)^2+O(x^(n+1))),n)) /* Ralf Stephan */

def a(n):
    if n==0: return 1
    return hypergeometric([1-n, n], [-n], 4).simplify()
[a(n) for n in range(24)] # Peter Luschny, Nov 30 2014

G.f.: (1+4*x*c(4*x)/3)/(1+x/3) = 1/(1-x*c(4*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = (1/n)*Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(4^m) = ((-1/3)^n)*(1 - 4*Sum_{k=0..n-1} C(k)*(-12)^k), n >= 1, a(0) = 1, with C(n) = A000108(n) (Catalan).
a(n) = Sum_{k=0...n} A059365(n, k)*4^(n-k). - Philippe Deléham, Jan 19 2004
D-finite with recurrence: 3*n*a(n) + (-47*n+72)*a(n-1) + 8*(-2*n+3)*a(n-2) = 0. - R. J. Mathar, Jun 07 2013 [verified by Georg Fischer, Jul 06 2021]
a(n) = hypergeometric([1-n, n], [-n], 4) for n > 0. - Peter Luschny, Nov 30 2014
a(n) ~ 2^(4*n + 2) / (49*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

A064088 Generalized Catalan numbers C(5; n).

Original entry on oeis.org

1, 1, 6, 61, 766, 10746, 161376, 2537781, 41260086, 687927166, 11698135396, 202104763026, 3537486504556, 62595852983236, 1117926476207316, 20124876291104421, 364797768048805926, 6652740911381353206, 121975721251036497636, 2247064873245590484966, 41573071647518070152196

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=5, beta =1 (or alpha=1, beta=5).

G. C. Greubel, Table of n, a(n) for n = 0..850

Cf. A064087 (C(4, n)).

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (9-Sqrt(1-20*x))/(2*(x+4)) )); // G. C. Greubel, May 02 2019

a[0] = 1; a[n_] := Sum[(n - m)*Binomial[n - 1 + m, m]*5^m/n, {m, 0, n - 1}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 09 2013 *)
CoefficientList[Series[(9-Sqrt[1-20*x])/(2*(x+4)), {x,0,30}], x] (* G. C. Greubel, May 02 2019 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-4*x^2)/(1+x)^2 +O(x^(n+1))), n)) /* Ralf Stephan */

( (9-sqrt(1-20*x))/(2*(x+4)) ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

G.f.: (1+5*x*c(5*x)/4)/(1+x/4) = 1/(1-x*c(5*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(5^m)/n, n >= 1, a(0) := 1.
a(n) = (-1/4)^n*(1 - 5*Sum_{k=0..n-1} C(k)*(-20)^k); with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*5^(n-k). - Philippe Deléham, Jan 19 2004
From Gary W. Adamson, Jul 18 2011: (Start)
a(n) = upper left term in M^n, M = an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  5, 5, 5, 0, 0, 0, ...
  5, 5, 5, 5, 0, 0, ...
  5, 5, 5, 5, 5, 0, ...
  5, 5, 5, 5, 5, 5, ...
  ... (End)
Conjecture: 4*n*a(n) +(-79*n+120)*a(n-1) +10*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013
a(n) ~ 4^n * 5^(n+1) / (81*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

A064089 Generalized Catalan numbers C(6; n).

Original entry on oeis.org

1, 1, 7, 85, 1279, 21517, 387607, 7312789, 142648495, 2853691357, 58226571271, 1207062556261, 25351452769567, 538285926177325, 11535690316148215, 249189167966657845, 5420206822556721295

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=6, beta =1 (or alpha=1, beta=6).

G. C. Greubel, Table of n, a(n) for n = 0..725

Cf. A064088 (C(5, n)).

Cf. A000108, A059365.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (11 -Sqrt(1-24*x))/(2*(x+5)) )); // G. C. Greubel, May 02 2019

CoefficientList[Series[(11 -Sqrt[1-24*x])/(2*(x+5)), {x, 0, 20}], x] (* G. C. Greubel, May 02 2019 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-5*x^2)/(1+x)^2+O(x^(n+1))), n)) /* Ralf Stephan */

my(x='x+O('x^20)); Vec((11 -sqrt(1-24*x))/(2*(x+5))) \\ G. C. Greubel, May 02 2019

((11 -sqrt(1-24*x))/(2*(x+5))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

G.f.: (1 + 6*x*c(6*x)/5)/(1+x/5) = 1/(1 - x*c(6*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(6^m)/n.
a(n) = (-1/5)^n*(1 - 6*Sum_{k=0..n-1} C(k)*(-30)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*6^(n-k). - Philippe Deléham, Jan 19 2004
Conjecture: 5*n*a(n) +(-119*n+180)*a(n-1) +12*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013
a(n) ~ 2^(3*n + 1) * 3^(n+1) / (121*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

A064090 Generalized Catalan numbers C(7; n).

Original entry on oeis.org

1, 1, 8, 113, 1982, 38886, 817062, 17981769, 409186310, 9549411950, 227307541448, 5497312072330, 134696099554276, 3336563455537768, 83419226227330722, 2102274863070771033, 53347639317495439302

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=7, beta =1 (or alpha=1, beta=7).

G. C. Greubel, Table of n, a(n) for n = 0..690

Cf. A064089 (C(6, n)).

Cf. A000108, A059365.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (13-Sqrt(1-28*x))/(2*(x+6)) )); // G. C. Greubel, May 02 2019

a[0]=1; a[n_]:= Sum[(n-m)*Binomial[n-1+m, m]*7^m/n, {m, 0, n-1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jul 09 2013 *)
CoefficientList[Series[(13 -Sqrt[1-28*x])/(2*(x+6)), {x, 0, 20}], x] (* G. C. Greubel, May 02 2019 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-6*x^2)/(1+x)^2 +O(x^(n+1))), n)) /* Ralf Stephan */

my(x='x+O('x^20)); Vec((13-sqrt(1-28*x))/(2*(x+6))) \\ G. C. Greubel, May 02 2019

((13-sqrt(1-28*x))/(2*(x+6))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

G.f.: (1+7*x*c(7*x)/6)/(1+x/6) = 1/(1-x*c(7*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(7^m)/n.
a(n) = (-1/6)^n*(1 - 7*Sum_{k=0..n-1} C(k)*(-42)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*7^(n-k). - Philippe Deléham, Jan 19 2004
Conjecture: 6*n*a(n) +(-167*n+252)*a(n-1) +14*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013
a(n) ~ 4^n * 7^(n+1) / (169*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

A064091 Generalized Catalan numbers C(8; n).

Original entry on oeis.org

1, 1, 9, 145, 2905, 65121, 1563561, 39322929, 1022586105, 27272680705, 741894295369, 20504949587409, 574176887116441, 16254518495907745, 464436319229036265, 13376293681432402545, 387925710986712480825

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1)= Y_{n}(n+1)= Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=8, beta =1 (or alpha=1, beta=8).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A064090 (C(7, n)).

Cf. A000108, A059365.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (15 - Sqrt(1-32*x))/(2*(x+7)) )); // G. C. Greubel, May 02 2019

a[0]=1; a[n_]:= Sum[(n-m)*Binomial[n+m-1, m]*(8^m)/n, {m, 0, n-1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jun 21 2013 *)
Table[FullSimplify[(-1)^(2*n)*2^(3+5*n)*(1/2*(2*n-1))! Hypergeometric2F1[1,1/2+n,2+n,-224]/(Sqrt[Pi]*(n+1)!)],{n,0,20}] (* Vaclav Kotesovec, Aug 13 2013 *)
CoefficientList[Series[(15 -Sqrt[1-32*x])/(2*(x+7)), {x,0,20}], x] (* G. C. Greubel, May 02 2019 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-7*x^2)/(1+x)^2+O(x^(n+1))), n)) /* Ralf Stephan */

my(x='x+O('x^20)); Vec((15 -sqrt(1-32*x))/(2*(x+7))) \\ G. C. Greubel, May 02 2019

((15 -sqrt(1-32*x))/(2*(x+7))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

G.f.: (1 + 8*x*c(8*x)/7)/(1+x/7) = 1/(1 - x*c(8*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(8^m)/n.
a(n) = (-1/7)^n*(1 - 8*Sum_{k=0..n-1} C(k)*(-56)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*8^(n-k). - Philippe Deléham, Jan 19 2004
Conjecture: 7*n*a(n) +(-223*n+336)*a(n-1) +16*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013
a(n) ~ 2^(5*n+3)/(225*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

A064092 Generalized Catalan numbers C(9; n).

Original entry on oeis.org

1, 1, 10, 181, 4078, 102826, 2777212, 78571837, 2298558934, 68964092542, 2110472708140, 65620725560578, 2067160250751436, 65833929303952564, 2116166898185821792, 68565914052628406221, 2237022199842087256678

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=9, beta =1 (or alpha=1, beta=9).

G. C. Greubel, Table of n, a(n) for n = 0..640

Cf. A064091 (C(8, n)).

Cf. A000108, A059365.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (17 - Sqrt(1-36*x))/(2*(x+8)) )); // G. C. Greubel, May 02 2019

a[0]=1; a[n_]:= Sum[(n-m)*Binomial[n-1+m, m]*9^m/n, {m, 0, n-1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jul 09 2013 *)
CoefficientList[Series[(17 -Sqrt[1-36*x])/(2*(x+8)), {x, 0, 20}], x] (* G. C. Greubel, May 02 2019 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-8*x^2)/(1+x)^2+O(x^(n+1))), n)) /* Ralf Stephan */

my(x='x+O('x^20)); Vec((17 -sqrt(1-36*x))/(2*(x+8))) \\ G. C. Greubel, May 02 2019

((17 -sqrt(1-36*x))/(2*(x+8))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

G.f.: (1 + 9*x*c(9*x)/8)/(1+x/8) = 1/(1 - x*c(9*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(9^m)/n.
a(n) = (-1/8)^n*(1 - 9*Sum_{k=0..n-1} C(k)*(-72)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*9^(n-k). - Philippe Deléham, Jan 19 2004
Conjecture: 8*n*a(n) +(-287*n+432)*a(n-1) +18*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013
a(n) ~ 4^n * 9^(n+1) / (289*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

A064093 Generalized Catalan numbers C(10; n).

Original entry on oeis.org

1, 1, 11, 221, 5531, 154941, 4649451, 146150061, 4750427771, 158361063581, 5384626548491, 186023930383501, 6511108452179611, 230400987949757821, 8228844334672249131, 296245683962814194541, 10739133812893020645051

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=10, beta =1 (or alpha=1, beta=10).

In general, for m>=1, C(m; n) ~ m * (4*m)^n / ((2*m - 1)^2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

G. C. Greubel, Table of n, a(n) for n = 0..620

Cf. A064092 (C(9, n)).

Cf. A000108, A059365.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (19 - Sqrt(1-40*x))/(2*(x+9)) )); // G. C. Greubel, May 02 2019

CoefficientList[Series[(19 -Sqrt[1-40*x])/(2*(x+9)), {x, 0, 20}], x] (* G. C. Greubel, May 02 2019 *)

my(x='x+O('x^20)); Vec((19 -sqrt(1-40*x))/(2*(x+9))) \\ G. C. Greubel, May 02 2019

((19 -sqrt(1-40*x))/(2*(x+9))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

G.f.: (1 + 10*x*c(10*x)/9)/(1+x/9) = 1/(1 - x*c(10*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(10^m)/n.
a(n) = (-1/9)^n*(1 - 10*Sum_{k=0..n-1} C(k)*(-90)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*10^(n-k). - Philippe Deléham, Jan 19 2004
a(n) ~ 2^(3*n + 1) * 5^(n+1) / (361*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

A090749 a(n) = 12 * C(2n+1,n-5) / (n+7).

Original entry on oeis.org

1, 12, 90, 544, 2907, 14364, 67298, 303600, 1332045, 5722860, 24192090, 100975680, 417225900, 1709984304, 6962078952, 28192122176, 113649492522, 456442180920, 1827459250276, 7297426411968, 29075683360185, 115631433392020, 459124809056550, 1820529677650320, 7210477496434485

Offset: 5

Philippe Deléham, Feb 15 2004

Also a diagonal of A059365 and A009766. See also A000108, A002057, A003517, A003518, A003519.

Number of standard tableaux of shape (n+6,n-5). - Emeric Deutsch, May 30 2004

G. C. Greubel, Table of n, a(n) for n = 5..1000
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.

Cf. A001622, A009766, A039598, A059365, A214292.

Cf. A000108, A002057, A003517, A003518, A003519.

Table[12*Binomial[2*n + 1, n - 5]/(n + 7), {n,5,50}] (* G. C. Greubel, Feb 07 2017 *)

for(n=5,50, print1(12*binomial(2*n+1,n-5)/(n+7), ", ")) \\ G. C. Greubel, Feb 07 2017

a(n) = A039598(n, 5) = A033184(n+7, 12).
G.f.: x^5*C(x)^12 with C(x) g.f. of A000108(Catalan).
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=11, a(n-6)=(-1)^(n-11)*coeff(charpoly(A,x),x^11). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n,n-6) for n > 5. - Reinhard Zumkeller, Jul 12 2012
From Karol A. Penson, Nov 21 2016: (Start)
O.g.f.: z^5 * 4^6/(1+sqrt(1-4*z))^12.
Recurrence: (-4*(n-5)^2-58*n+80)*a(n+1)-(-n^2-6*n+27)*a(n+2)=0, a(0),a(1),a(2),a(3),a(4)=0,a(5)=1,a(6)=12, n>=5.
Asymptotics: (-903+24*n)*4^n*sqrt(1/n)/(sqrt(Pi)*n^2).
Integral representation as n-th moment of a signed function W(x) on x=(0,4),in Maple notation: a(n+5)=int(x^n*W(x),x=0..4),n=0,1,..., where W(x)=(256/231)*sqrt(4-x)*JacobiP(5, 1/2, 1/2, (1/2)*x-1)*x^(11/2)/Pi and JacobiP are Jacobi polynomials. Note that W(0)=W(4)=0. (End).
From Ilya Gutkovskiy, Nov 21 2016: (Start)
E.g.f.: 6*exp(2*x)*BesselI(6,2*x)/x.
a(n) ~ 3*2^(2*n+3)/(sqrt(Pi)*n^(3/2)). (End)
From Amiram Eldar, Sep 26 2022: (Start)
Sum_{n>=5} 1/a(n) = 88699/15120 - 71*Pi/(27*sqrt(3)).
Sum_{n>=5} (-1)^(n+1)/a(n) = 102638*log(phi)/(75*sqrt(5)) - 22194839/75600, where phi is the golden ratio (A001622). (End)

Missing term 113649492522 inserted by Ilya Gutkovskiy, Dec 07 2016

cp's OEIS Frontend

A099039 Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.

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A064087 Generalized Catalan numbers C(4; n).

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A064088 Generalized Catalan numbers C(5; n).

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A064089 Generalized Catalan numbers C(6; n).

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A064090 Generalized Catalan numbers C(7; n).

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A064091 Generalized Catalan numbers C(8; n).

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A064092 Generalized Catalan numbers C(9; n).