A062992 -id:A062992 - cp's OEIS frontend

A115195 Triangle of numbers, called Y(1,2), related to generalized Catalan numbers A062992(n) = C(2;n+1) = A064062(n+1).

1, 2, 3, 4, 10, 13, 8, 28, 54, 67, 16, 72, 180, 314, 381, 32, 176, 536, 1164, 1926, 2307, 64, 416, 1488, 3816, 7668, 12282, 14589, 128, 960, 3936, 11568, 26904, 51468, 80646, 95235, 256, 2176, 10048, 33184, 86992, 189928, 351220, 541690, 636925, 512, 4864

Offset: 0

Wolfdieter Lang, Feb 23 2006

This triangle Y(1,2) appears in the totally asymmetric exclusion process for the (unphysical) values alpha=1, beta=2. See the Derrida et al. refs. given under A064094, where the triangle entries are called Y_{N,K} for given alpha and beta.
The main diagonal (M=1) gives the generalized Catalan sequence C(2,n+1):=A064062(n+1).
The diagonal sequences give A064062(n+1), 2*A084076, 4*A115194, 8*A115202, 16*A115203, 32*A115204 for n+1>= M=1,..,6.

			Triangle begins:
   1;
   2,  3;
   4, 10,  13;
   8, 28,  54,  67;
  16, 72, 180, 314, 381;
  ...

B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
Wolfdieter Lang, First 10 rows.

Row sums give A084076.

G.f. m-th diagonal, m>=1: ((1 + 2*x*c(2*x))*(2*x*c(2*x))^m)/(2*x*(1+x)) with c(x) the o.g.f. of A000108 (Catalan).

A138392 Hankel transform of A062992 with interpolated zeros.

Original entry on oeis.org

1, 3, 12, 128, 2048, 81920, 5242880, 805306368, 206158430208, 123145302310912, 126100789566373888, 295147905179352825856, 1208925819614629174706176, 11141460353568422474092118016, 182541686432865033815525261574144

Offset: 0

Paul Barry, Mar 18 2008

A138392 := proc(n)
    2^(binomial(n+1,2)+0^n-1)*(2+floor((n+1)/2)-0^n) ;
end proc:
seq(%(n),n=0..13) ; # R. J. Mathar, Feb 23 2015

a(n)=2^(comb(n+1,2)+0^n-1)*(2+floor((n+1)/2)-0^n)

A064062 Generalized Catalan numbers C(2; n).

Original entry on oeis.org

1, 1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 210731011, 1493303293, 10678370307, 76957679613, 558403682307, 4075996839933, 29909606989827, 220510631755773, 1632599134961667, 12133359132082173

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=2, beta=1 (or alpha=1, beta=2).
a(n) = number of Dyck n-paths (A000108) in which each upstep (U) not at ground level is colored red (R) or blue (B). For example, a(3)=3 counts URDD, UBDD, UDUD (D=downstep). - David Callan, Mar 30 2007
The Hankel transform of this sequence is A002416. - Philippe Deléham, Nov 19 2007
The sequence a(n)/2^n, with g.f. 1/(1-xc(x)/2), has Hankel transform 1/2^n. - Paul Barry, Apr 14 2008
The REVERT transform of the odd numbers [1,3,5,7,9,...] is [1, -3, 13, -67, 381, -2307, 14589, -95235, 636925, ...] - N. J. A. Sloane, May 26 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Abate and W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, example section 3.
Jean-Luc Baril, Sergey Kirgizov, and Mehdi Naima, A lattice on Dyck paths close to the Tamari lattice, arXiv:2309.00426 [math.CO], 2023.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From _N. J. A. Sloane_, Oct 08 2012
J. Bloom and S. Elizalde, Pattern avoidance in matchings and partitions, arXiv:1211.3442 [math.CO] (2012) Theorem 6.1.
N. Bonichon, C. Gavoille and N. Hanusse, Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation, In Proceedings of WG'03, volume 2880 of LNCS, pp. 81-92, 2003.
Alexander Burstein, Sergi Elizalde and Toufik Mansour, Restricted Dumont permutations, Dyck paths and noncrossing partitions, arXiv:math/0610234 [math.CO], 2006.
Xiang-Ke Chang, X.-B. Hu, H. Lei and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).
Simone Fioravanti, Steve Hanneke, Shay Moran, Hilla Schefler, and Iska Tsubari, Ramsey Theorems for Trees and a General 'Private Learning Implies Online Learning' Theorem, arXiv:2407.07765 [cs.LG], 2024. See p. 44.
Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics 24(2) (2017), #P2.3.
N. J. A. Sloane, Transforms
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011-2012.

Generalized Catalan numbers C(m; n): A000012 (m = 0), A000108 (m = 1), A064063 (m = 3) and A064087 - A064093 (m = 4 thru 10); A064310 (m = -1), A064311 (m = -2) and A064325 - A064333 (m = -3 thru -11).

Cf. A064334, A119529.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (3 - Sqrt(1-8*x))/(2*(1+x)) )); // G. C. Greubel, Sep 27 2024

1, seq(simplify(hypergeom([1-n,n],[-n],2)), n=1..100); # Robert Israel, Nov 30 2014

a[0]=1; a[1]=1; a[n_]/;n>=2 := a[n] = a[n-1] + Sum[(a[k] + a[k-1])a[n-k],{k,n-1}]; Table[a[n],{n,0,10}] (* David Callan, Aug 27 2009 *)
a[n_] := 2*Sum[ (-1)^j*2^(n-j-1)*Binomial[2*(n-j-1), n-j-1]/(n-j), {j, 0, n-1}] + (-1)^n; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 03 2013 *)

{a(n)=polcoeff((3-sqrt(1-8*x+x*O(x^n)))/(2+2*x),n)}

{a(n)=local(A=1+x); for(i=1, n, A=1+A^4*intformal(1/(A^2+x*O(x^n)))); polcoeff(A, n)} \\ Paul D. Hanna, Dec 24 2013
for(n=0, 25, print1(a(n), ", "))

{a(n)=polcoeff(1/(1 - serreverse(x-2*x^2 +x^2*O(x^n))),n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Nov 30 2014

def a(n):
    if n==0: return 1
    return hypergeometric([1-n, n], [-n], 2).simplify()
[a(n) for n in range(22)] # Peter Luschny, Dec 01 2014

G.f.: (1 + 2*x*C(2*x)) / (1+x) = 1/(1 - x*C(2*x)) with C(x) g.f. of Catalan numbers A000108.
a(n) = A062992(n-1) = Sum_{m = 0..n-1} (n-m)*binomial(n-1+m, m)*(2^m)/n, n >= 1, a(0) = 1.
a(n) = Sum_{k = 0..n} A059365(n, k)*2^(n-k). - Philippe Deléham, Jan 19 2004
G.f.: 1/(1-x/(1-2x/(1-2x/(1-2x/(1-.... = 1/(1-x-2x^2/(1-4x-4x^2/(1-4x-4x^2/(1-.... (continued fractions). - Paul Barry, Jan 30 2009
a(n) = (32/Pi)*Integral_{x = 0..1} (8*x)^(n-1)*sqrt(x*(1-x)) / (8*x+1). - Groux Roland, Dec 12 2010
a(n+2) = 8^(n+2)*( c(n+2)-c(1)*c(n+1) - Sum_{i=0..n-1} 8^(-i-2)*c(n-i)*a(i+2) ) with c(n) = Catalan(n+2)/2^(2*n+1). - Groux Roland, Dec 12 2010
a(n) = the upper left term in M^n, M = the production matrix:
  1, 1
  2, 2, 1
  4, 4, 2, 1
  8, 8, 4, 2, 1
  ... - Gary W. Adamson, Jul 08 2011
D-finite with recurrence: n*a(n) + (12-7n)*a(n-1) + 4*(3-2n)*a(n-2) = 0. - R. J. Mathar, Nov 16 2011 (This follows easily from the generating function. - Robert Israel, Nov 30 2014)
G.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^2 dx. - Paul D. Hanna, Dec 24 2013
G.f. satisfies: Integral 1/A(x)^2 dx = x - x^2*G(x), where G(x) is the o.g.f. of A000257, the number of rooted bicubic maps. - Paul D. Hanna, Dec 24 2013
G.f. A(x) satisfies: A(x - 2*x^2) = 1/(1-x). - Paul D. Hanna, Nov 30 2014
a(n) = hypergeometric([1-n, n], [-n], 2) for n > 0. - Peter Luschny, Nov 30 2014
G.f.: (3 - sqrt(1-8*x))/(2*(x+1)). - Robert Israel, Nov 30 2014
a(n) ~ 2^(3*n+1) / (9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 22 2014
O.g.f. A(x) =  1 + series reversion of (x*(1 - x)/(1 + x)^2). Logarithmically differentiating (A(x) - 1)/x gives 3 + 17*x + 111*x^2 + ..., essentially a g.f for A119259. - Peter Bala, Oct 01 2015
From Peter Bala, Jan 06 2022: (Start)
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + ... is a g.f. for A022558.
The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)

A062991 Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x).

Original entry on oeis.org

1, 2, -1, 5, -6, 2, 14, -28, 20, -5, 42, -120, 135, -70, 14, 132, -495, 770, -616, 252, -42, 429, -2002, 4004, -4368, 2730, -924, 132, 1430, -8008, 19656, -27300, 23100, -11880, 3432, -429, 4862, -31824, 92820, -157080, 168300, -116688, 51051, -12870, 1430

Offset: 0

Wolfdieter Lang, Jul 12 2001

The g.f. for the sequence of column m of triangle A009766(n,m) (or Catalan A033184(n,n-m) diagonals) is N(2; m-1,x)*(x^m)/(1-x)^(m+1), m >= 1, with N(2; n,x) = Sum_{k=0..n} T(n,k)*x^k.
For k=0..1 the column sequences give A000108(n+1) (Catalan), -A002694. The row sums give A000012 (powers of 1) and (unsigned) A062992.
Another version of [1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [0, -1, -1, -1, -1, -1, -1, -1, ...] = 1; 1, 0; 2, -1, 0; 5, -6, 2, 0; 14, -28, 20, -5, 0; 42, -120, 135, -70, 14, 0; ... where DELTA is Deléham's operator defined in A084938.
The positive triangle is |T(n,k)| = binomial(2*n+2, n-k)*binomial(n+k, k)/(n+1). - Paul Barry, May 11 2005

			The triangle N2 = {a(n,k)} begins:
n\k      0       1      2       3       4       5      6       7     8     9
----------------------------------------------------------------------------
0:       1
1:       2      -1
2:       5      -6      2
3:      14     -28     20      -5
4:      42    -120    135     -70      14
5:     132    -495    770    -616     252     -42
6:     429   -2002   4004   -4368    2730    -924    132
7:    1430   -8008  19656  -27300   23100  -11880   3432    -429
8:    4862  -31824  92820 -157080  168300 -116688  51051  -12870  1430
9:   16796 -125970 426360 -852720 1108536 -969969 570570 -217360 48620 -4862
... formatted by _Wolfdieter Lang_, Jan 20 2020
N(2; 2, x)= 5 - 6*x + 2*x^2.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
C. A. Francisco, J. Mermin, and J. Schweig, Catalan numbers, binary trees, and pointed pseudotriangulations, 2013.
V. E. Hoggatt Jr. and Marjorie Bicknell-Johnson, Numerator Polynomial Coefficient Arrays for Catalan and Related Sequence Convolution Triangles, The Fibonacci Quarterly 15 (1977) 30-34. [On p. 31, in the line n = 1, 14 is missing in S_1^4. - _Wolfdieter Lang_, Jan 20 2020 ]
Joseph T. Iosue, Adam Ehrenberg, Dominik Hangleiter, Abhinav Deshpande, and Alexey V. Gorshkov, Page curves and typical entanglement in linear optics, arXiv:2209.06838 [quant-ph], 2022.
A. Lakshminarayan, Z. Puchala, and K. Zyczkowski, Diagonal unitary entangling gates and contradiagonal quantum states, arXiv preprint arXiv:1407.1169 [quant-ph], 2014.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A000108, A002694, A009766, A033184, A062992, A084938, A089434, A094385.
For an unsigned version see Borel's triangle, A234950.
Sums include: A000012 (row), A000079 (diagonal), A064062 (signed row), A071356 (signed diagonal).

A062991:= func< n,k | (-1)^k*Binomial(2*n+2,n-k)*Binomial(n+k,k)/(n+1) >;
[A062991(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 27 2024

T[n_, k_] := 2 (-1)^k Binomial[2n+1, n] (n-k+1) Binomial[n+1, k]/((k+n+1)(k+n+2));
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2018 *)

def A062991(n,k): return (-1)^k*binomial(2*n+2,n-k)*binomial(n+k,k)/(n+1)
flatten([[A062991(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 27 2024

T(n, k) = [x^k] N(2; n, x) with N(2; n, x) = (N(2; n-1, x) - A000108(n)*(1-x)^(n+1))/x, N(2; 0, x) = 1.
T(n, k) = T(n-1, k+1) + (-1)^k*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=0, .., (n-2); T(n, k) = (-1)^k*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=(n-1) or n; else 0.
O.g.f. (with offset 1) is the series reversion w.r.t. x of x*(1+x*t)/(1+x)^2. If R(n,t) denotes the n-th row polynomial of this triangle then R(n,1-t) is the n-th row polynomial of A009766. Cf. A089434. - Peter Bala, Jul 15 2012
From G. C. Greubel, Sep 27 2024: (Start)
Sum_{k=0..n} T(n, k) = A000012(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A064062(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000079(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A071356(n). (End)

A107111 Number array whose rows are the series reversions of x(1-x)/(1+x)^k, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 13, 22, 14, 1, 5, 23, 67, 90, 42, 1, 6, 36, 156, 381, 394, 132, 1, 7, 52, 305, 1162, 2307, 1806, 429, 1, 8, 71, 530, 2833, 9192, 14589, 8558, 1430, 1, 9, 93, 847, 5919, 27916, 75819, 95235, 41586, 4862, 1, 10, 118, 1272, 11070, 70098, 286632, 644908, 636925, 206098, 16796

Offset: 0

Paul Barry, May 12 2005

First row is the Catalan numbers A000108, second row is the large Schroeder numbers A006318, third row is A062992, fourth row is A007297. As a number triangle, this is T(n,k)=if(k<=n,sum{j=0..k, binomial((n-k)(k+1),k-j)*binomial(k+j,j)}/(k+1),0) with row sums A107112 and diagonal sums A107113.

			Array begins
1,1,2,5,14,42,132,...
1,2,6,22,90,394,1806,...
1,3,13,67,381,2307,14589,...
1,4,23,156,1162,9192,75819,...

Cf. A366012.

A107111 := proc(n,k)
    add(binomial(n*(k+1),k-j)*binomial(k+j,j),j=0..k);
    %/(k+1) ;
end proc: # R. J. Mathar, Aug 02 2016

T[n_, k_] := Sum[Binomial[n (k + 1), k - j] Binomial[k + j, j], {j, 0, k}]/(k + 1);
Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2020 *)

T(n, k)=sum{j=0..k, binomial(n(k+1), k-j)*binomial(k+j, j)}/(k+1)

A234950 Borel's triangle read by rows: T(n,k) = Sum_{s=k..n} binomial(s,k)*C(n,s), where C(n,s) is an entry in Catalan's triangle A009766.

Original entry on oeis.org

1, 2, 1, 5, 6, 2, 14, 28, 20, 5, 42, 120, 135, 70, 14, 132, 495, 770, 616, 252, 42, 429, 2002, 4004, 4368, 2730, 924, 132, 1430, 8008, 19656, 27300, 23100, 11880, 3432, 429, 4862, 31824, 92820, 157080, 168300, 116688, 51051, 12870, 1430

Offset: 0

N. J. A. Sloane, Jan 11 2014

			Triangle begins:
     1,
     2,    1,
     5,    6,     2,
    14,   28,    20,     5,
    42,  120,   135,    70,    14,
   132,  495,   770,   616,   252,    42,
   429, 2002,  4004,  4368,  2730,   924,  132,
  1430, 8008, 19656, 27300, 23100, 11880, 3432, 429,
  ...

Reinhard Zumkeller, Rows n=0..125 of triangle, flattened
Antoine Abram, Florent Hivert, James D. Mitchell, Jean-Christophe Novelli, and Maria Tsalakou, Power Quotients of Plactic-like Monoids, arXiv:2406.16387 [math.CO], 2024. See p. 5.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Steve Butler, R. Graham, and C. H. Yan, Parking distributions on trees, European Journal of Combinatorics 65 (2017), 168-185.
Yue Cai and Catherine Yan, Counting with Borel's triangle, Texas A&M University.
Yue Cai and Catherine Yan, Counting with Borel's triangle, arXiv:1804.01597 [math.CO], 2018.
G. Chatel and V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.
C. A. Francisco, J. Mermin, and J. Schweig, Catalan numbers, binary trees, and pointed pseudotriangulations, preprint 2013; European Journal of Combinatorics, Volume 45, April 2015, pp. 85-96.
Lord C. Kavi and Michael W. Newman, Counting closed walks in infinite regular trees using Catalan and Borel's triangles, arXiv:2212.08795 [math.CO], 2022.
A. Lakshminarayan, Z. Puchala, and K. Zyczkowski, Diagonal unitary entangling gates and contradiagonal quantum states, arXiv preprint arXiv:1407.1169 [quant-ph], 2014.
Jeffrey B. Remmel, Consecutive Up-down Patterns in Up-down Permutations, Electron. J. Combin., 21 (2014), #P3.2. See pp. 21-22. - _N. J. A. Sloane_, Jul 12 2014

A062991 is a signed version. See also A094385 for another version.
Cf. A009766.
The two borders give the Catalan numbers A000108.
Cf. A062992 (row sums).
The second and third columns give A002694 and A244887.

a234950 n k = sum [a007318 s k * a009766 n s | s <- [k..n]]
a234950_row n = map (a234950 n) [0..n]
a234950_tabl = map a234950_row [0..]
-- Reinhard Zumkeller, Jan 12 2014

T := (n,k) -> 2*binomial(2*n+1,n)*(n-k+1)*binomial(n+1,k)/((k+n+1)*(k+n+2)):
seq(seq(T(n,k), k=0..n), n=0..8); # Peter Luschny, Sep 04 2018

T[n_, k_] := 2 Binomial[2n+1, n] (n-k+1) Binomial[n+1, k]/((k+n+1)(k+n+2));
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2018, from Maple *)

T(n,k) = sum(s=k, n, binomial(s, k)*binomial(n+s, n)*(n-s+1)/(n+1));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print();); \\ Michel Marcus, Sep 06 2015

G.f.: 1/x*(1-sqrt(1-4*x-4*x*y))/(1+2*y+sqrt(1-4*x-4*x*y)). - Vladimir Kruchinin, Sep 04 2018

T(n,k) = 2*binomial(2*n+1,n)*(n-k+1)*binomial(n+1,k)/((k+n+1)*(k+n+2)). - Peter Luschny, Sep 04 2018

A114191 Expansion of 1/(1+xc(-2x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, -1, 3, -13, 67, -381, 2307, -14589, 95235, -636925, 4341763, -30056445, 210731011, -1493303293, 10678370307, -76957679613, 558403682307, -4075996839933, 29909606989827, -220510631755773, 1632599134961667, -12133359132082173, 90485602494971907, -676925762716041213

Offset: 0

Paul Barry, Nov 16 2005

First column of A114189. Row sums of A114193. Alternating sign version of A062992.

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

CoefficientList[Series[4/(3+Sqrt[1+8*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

G.f.: 4/(3+sqrt(1+8*x)).
a(n) = Sum_{k=0..n} (-2)^(n-k)*A039599(n, k) = Sum_{k=0..n} (-2)^(n-k)*C(2*n, n-k)*(2*k+1)/(n+k+1). - Philippe Deléham, Nov 17 2005
Conjecture: n*a(n) + (7*n-12)*a(n-1) + 4*(3-2*n)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
a(n) ~ (-1)^n * 2^(3*n+1) / (9 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2014

A115137 Second diagonal of triangle A113647 (called Y(2,1)).

Original entry on oeis.org

1, 1, 7, 41, 247, 1545, 9975, 66057, 446455, 3067913, 21372919, 150618121, 1071841271, 7691763721, 55600938999, 404488323081, 2959189475319, 21757613309961, 160691417776119, 1191577871450121, 8868160862158839

Offset: 0

Wolfdieter Lang, Jan 13 2006

			41=a(3)= A062992(3) - 2*A062992(2) = 67 - 2*13.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

CoefficientList[Series[(1-2*x)*(2*(1-Sqrt[1-8*x])/(4*x)-1)/(1+x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

a(n)= b(n) - 2*b(n-1) with b(n):=A062992(n)= A064062(n+1), n>=1. a(0):=1.
G.f.: (1-2*x)*(2*c(2*x)-1)/(1+x) with c(x) g.f. of A000108 (Catalan).
a(n)= A113647(n, n), n>=1.
Recurrence: (n-2)*(n+1)*a(n) = (7*n^2-19*n+14)*a(n-1) + 4*(n-1)*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(3*n+2)/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

A115125 A sequence related to Catalan numbers A000108.

Original entry on oeis.org

1, 2, 4, 16, 80, 448, 2688, 16896, 109824, 732160, 4978688, 34398208, 240787456, 1704034304, 12171673600, 87636049920, 635361361920, 4634400522240, 33985603829760, 250420238745600, 1853109766717440, 13765958267043840, 102618961627054080, 767411365211013120

Offset: 0

Wolfdieter Lang, Jan 13 2006

Essentially identical to A025225.
The convolution of this sequence with the sequence {(-1)^n} is A064062 (see also A062992).
The sequence A064062 appears in the Derrida et al. 1992 reference (see A064094) for alpha=2, beta=1 (or alpha=1, beta=2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.

Cf. A000108, A025225, A062992, A064062, A064094.

[1] cat [2^n*Binomial(2*n-2, n-1)/n: n in [1..30]]; // G. C. Greubel, May 03 2018

a:= n-> `if`(n=0, 1, 2^n*binomial(2*n-2, n-1)/n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2022

a[0] = 1; a[n_] := 2^n*CatalanNumber[n - 1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 09 2013 *)

a(n)=if(n==0,1,polcoeff((1-sqrt(1-8*x+x*O(x^n)))/2,n)); \\ Joerg Arndt, May 14 2013

a(n) = C(n-1)*2^n, n>=1, a(0):=1, with C(n):=A000108(n) (Catalan).
G.f.: 1 + (2*x)*c(2*x) with c(x):=(1-sqrt(1-4*x))/(2*x), the o.g.f. of Catalan numbers A000108.
a(n) = A025225(n), n>0. - R. J. Mathar, Aug 11 2008
G.f.: (3 - sqrt(1-8*x))/2 = 2 - U(0) where U(k)=1 - 2*x/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 29 2012
G.f.: 2 - 1/Q(0), where Q(k)= 1 + (8*k+2)*x/(k+1 - x*(2*k+2)*(8*k+6)/(2*x*(8*k+6) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013

A366012 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(n*(n+1),n-k).

Original entry on oeis.org

1, 2, 13, 156, 2833, 70098, 2214280, 85464984, 3906724321, 206648387550, 12425282899588, 837384222603448, 62539219710804627, 5127758187193514824, 457986530357734020432, 44263628968974498793648, 4602969726808566383149761, 512486177498084438210961270, 60827938291895363867587959628

Offset: 0

Ilya Gutkovskiy, Sep 25 2023

nonn

Main diagonal of A107111.

Cf. A000108, A006318, A062992, A135861, A263843, A365754, A365755.

Table[1/(n + 1) Sum[Binomial[n + k, n] Binomial[n (n + 1), n - k], {k, 0, n}], {n, 0, 18}]
Table[SeriesCoefficient[(1/x) InverseSeries[Series[x (1 - x)/(1 + x)^n, {x, 0, n + 1}], x], {x, 0, n}], {n, 0, 18}]

a(n) = [x^n] (1/x) * Series_Reversion( x * (1 - x) / (1 + x)^n ).

a(n) ~ exp(n + 3/2) * n^(n - 3/2) / sqrt(2*Pi). - Vaclav Kotesovec, Sep 26 2023

cp's OEIS Frontend

A115195 Triangle of numbers, called Y(1,2), related to generalized Catalan numbers A062992(n) = C(2;n+1) = A064062(n+1).

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A138392 Hankel transform of A062992 with interpolated zeros.

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A064062 Generalized Catalan numbers C(2; n).

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A062991 Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x).

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A107111 Number array whose rows are the series reversions of x(1-x)/(1+x)^k, read by antidiagonals.

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A234950 Borel's triangle read by rows: T(n,k) = Sum_{s=k..n} binomial(s,k)*C(n,s), where C(n,s) is an entry in Catalan's triangle A009766.

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A114191 Expansion of 1/(1+x*c(-2*x)), c(x) the g.f. of A000108.

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A115137 Second diagonal of triangle A113647 (called Y(2,1)).

A114191 Expansion of 1/(1+xc(-2x)), c(x) the g.f. of A000108.