A064087 -id:A064087 - cp's OEIS frontend

A106566 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, ... ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ... ] where DELTA is the operator defined in A084938.

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

Offset: 0

Philippe Deléham, May 30 2005

Catalan convolution triangle; g.f. for column k: (x*c(x))^k with c(x) g.f. for A000108 (Catalan numbers).
Riordan array (1, xc(x)), where c(x) the g.f. of A000108; inverse of Riordan array (1, x*(1-x)) (see A109466).
Diagonal sums give A132364. - Philippe Deléham, Nov 11 2007

			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;
From _Paul Barry_, Sep 28 2009: (Start)
Production array 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 (End)

Alois P. Heinz, Rows n = 0..140, flattened
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
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.
FindStat - Combinatorial Statistic Finder, The number of touch points of a Dyck path, The number of initial rises of a Dyck paths, The number of nodes on the left branch of the tree, The number of subtrees.
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, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.

Column k for k = 0, 1, 2, ..., 13: A000007, A000108, A000108, A000245, A002057, A000344, A003517, A000588, A003517, A001392, A003518, A000589, A003519, A000590
The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.
Diagonals: A000012, A001477, A000096, A005586, A005587, A005557, A064059, A064061
See also A009766, A033184, A059365 for other versions.
Generalized Catalan numbers C(x, n) for -11 <= x <= 10: A064333, A064332, A064331, A064330, A064329, A064328, A064327, A064326, A064325, A064311, A064310, A000012, A000108, A064062, A064063, A064087, A064088, A064089, A064090, A064091, A064092, A064093.
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, ...

A106566:= func< n,k | n eq 0 select 1 else (k/n)*Binomial(2*n-k-1, n-k) >;
[A106566(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 06 2021

A106566 := proc(n,k)
    if n = 0 then
        1;
    elif k < 0 or k > n then
        0;
    else
        binomial(2*n-k-1,n-k)*k/n ;
    end if;
end proc: # R. J. Mathar, Mar 01 2015

T[n_, k_] := Binomial[2n-k-1, n-k]*k/n; T[0, 0] = 1; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2017 *)
(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, #(1-Sqrt[1-4#])/(2#)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

{T(n, k) = if( k<=0 || k>n, n==0 && k==0, binomial(2*n - k, n) * k/(2*n - k))}; /* Michael Somos, Oct 01 2022 */

def A106566(n, k): return 1 if (n==0) else (k/n)*binomial(2*n-k-1, n-k)
flatten([[A106566(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 06 2021

T(n, k) = binomial(2n-k-1, n-k)*k/n for 0 <= k <= n with n > 0; T(0, 0) = 1; T(0, k) = 0 if k > 0.
T(0, 0) = 1; T(n, 0) = 0 if n > 0; T(0, k) = 0 if k > 0; for k > 0 and n > 0: T(n, k) = Sum_{j>=0} T(n-1, k-1+j).
Sum_{j>=0} T(n+j, 2j) = binomial(2n-1, n), n > 0.
Sum_{j>=0} T(n+j, 2j+1) = binomial(2n-2, n-1), n > 0.
Sum_{k>=0} (-1)^(n+k)*T(n, k) = A064310(n). T(n, k) = (-1)^(n+k)*A099039(n, k).
Sum_{k=0..n} T(n, k)*x^k = A000007(n), A000108(n), A000984(n), A007854(n), A076035(n), A076036(n), A127628(n), A126694(n), A115970(n) for x = 0,1,2,3,4,5,6,7,8 respectively.
Sum_{k>=0} T(n, k)*x^(n-k) = C(x, n); C(x, n) are the generalized Catalan numbers.
Sum_{j=0..n-k} T(n+k,2*k+j) = A039599(n,k).
Sum_{j>=0} T(n,j)*binomial(j,k) = A039599(n,k).
Sum_{k=0..n} T(n,k)*A000108(k) = A127632(n).
Sum_{k=0..n} T(n,k)*(x+1)^k*x^(n-k) = A000012(n), A000984(n), A089022(n), A035610(n), A130976(n), A130977(n), A130978(n), A130979(n), A130980(n), A131521(n) for x= 0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Aug 25 2007
Sum_{k=0..n} T(n,k)*A000108(k-1) = A121988(n), with A000108(-1)=0. - Philippe Deléham, Aug 27 2007
Sum_{k=0..n} T(n,k)*(-x)^k = A000007(n), A126983(n), A126984(n), A126982(n), A126986(n), A126987(n), A127017(n), A127016(n), A126985(n), A127053(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Oct 27 2007
T(n,k)*2^(n-k) = A110510(n,k); T(n,k)*3^(n-k) = A110518(n,k). - Philippe Deléham, Nov 11 2007
Sum_{k=0..n} T(n,k)*A000045(k) = A109262(n), A000045: Fibonacci numbers. - Philippe Deléham, Oct 28 2008
Sum_{k=0..n} T(n,k)*A000129(k) = A143464(n), A000129: Pell numbers. - Philippe Deléham, Oct 28 2008
Sum_{k=0..n} T(n,k)*A100335(k) = A002450(n). - Philippe Deléham, Oct 30 2008
Sum_{k=0..n} T(n,k)*A100334(k) = A001906(n). - Philippe Deléham, Oct 30 2008
Sum_{k=0..n} T(n,k)*A099322(k) = A015565(n). - Philippe Deléham, Oct 30 2008
Sum_{k=0..n} T(n,k)*A106233(k) = A003462(n). - Philippe Deléham, Oct 30 2008
Sum_{k=0..n} T(n,k)*A151821(k+1) = A100320(n). - Philippe Deléham, Oct 30 2008
Sum_{k=0..n} T(n,k)*A082505(k+1) = A144706(n). - Philippe Deléham, Oct 30 2008
Sum_{k=0..n} T(n,k)*A000045(2k+2) = A026671(n). - Philippe Deléham, Feb 11 2009
Sum_{k=0..n} T(n,k)*A122367(k) = A026726(n). - Philippe Deléham, Feb 11 2009
Sum_{k=0..n} T(n,k)*A008619(k) = A000958(n+1). - Philippe Deléham, Nov 15 2009
Sum_{k=0..n} T(n,k)*A027941(k+1) = A026674(n+1). - Philippe Deléham, Feb 01 2014
G.f.: Sum_{n>=0, k>=0} T(n, k)*x^k*z^n = 1/(1 - x*z*c(z)) where c(z) the g.f. of A000108. - Michael Somos, Oct 01 2022

Formula corrected by Philippe Deléham, Oct 31 2008
Corrected by Philippe Deléham, Sep 17 2009
Corrected by Alois P. Heinz, Aug 02 2012

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)

A064094 Triangle composed of generalized Catalan numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 13, 4, 1, 1, 1, 42, 67, 25, 5, 1, 1, 1, 132, 381, 190, 41, 6, 1, 1, 1, 429, 2307, 1606, 413, 61, 7, 1, 1, 1, 1430, 14589, 14506, 4641, 766, 85, 8, 1, 1, 1, 4862, 95235, 137089, 55797, 10746, 1279, 113, 9, 1, 1

Offset: 0

Wolfdieter Lang, Sep 13 2001

The column m sequence (without leading zeros and the first 1) appears in the Derrida et al. 1992 reference as Z_{N}=Y_{N}(N+1), N >=0, for alpha = m, beta = 1 (or alpha = 1, beta = m). In the Derrida et al. 1993 reference the formula in eq. (39) gives Z_{N}(alpha,beta)/(alpha*beta)^N for N>=1.

			Triangle begins:
  1;
  1,    1;
  1,    1,     1;
  1,    2,     1,     1;
  1,    5,     3,     1,    1;
  1,   14,    13,     4,    1,   1;
  1,   42,    67,    25,    5,   1,   1;
  1,  132,   381,   190,   41,   6,   1,   1;
  1,  429,  2307,  1606,  413,  61,   7,   1,   1;
  1, 1430, 14589, 14506, 4641, 766,  85,   8,   1,   1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
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.
B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) p. 1513.

Diagonals: A000012, A000012, A000027, A001844, A064096, A064302, A064303, A064304, A064305.
Columns (without leading zeros): A000012 (k=0), A000108 (k=1), A064062 (k=2), A064063 (k=3), A064087 (k=4), A064088 (k=5), A064089 (k=6), A064090 (k=7), A064091 (k=8), A064092 (k=9), A064093 (k=10).
Cf. A064095 (row sums).

function A064094(n,k)
  if k eq 0 or k eq n then return 1;
  else return (&+[(n-k-j)*Binomial(n-k-1+j, j)*k^j: j in [0..n-k-1]])/(n-k);
  end if;
end function;
[A064094(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 27 2024

T[n_, 0] = 1; T[n_, 1] := CatalanNumber[n - 1]; T[n_, n_] = 1; T[n_, m_] := (1/(1 - m))^(n - m)*(1 - m*Sum[ CatalanNumber[k]*(m*(1 - m))^k, {k, 0, n - m - 1}]); Table[ T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013 *)

def A064094(n,k):
    if (k==0 or k==n): return 1
    else: return sum((n-k-j)*binomial(n-k-1+j,j)*k^j for j in range(n-k))//(n-k)
flatten([[A064094(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 27 2024

G.f. for column m: (x^m)/(1-x*c(m*x)) = (x^m)*((m-1)+m*x*c(m*x))/(m-1+x) with the g.f. c(x) of Catalan numbers A000108.

T(n, m) = Sum_{j=0..n-m-1} (n-m-j)*binomial(n-m-1+j, j)*(m^j)/(n-m) or T(n, m) = (1/(1-m))^(n-m)*(1 - m*Sum_{j=0..n-m-1} C(j)*(m*(1-m))^j ), for n - m >= 1, T(n, n) = 1, T(n, m) = 0 if nA000108(k) (Catalan).

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

A094385 Triangle read by rows: T(n, k) = binomial(2n, k-1)binomial(2*n-k-1, n-k)/n for n, k >= 1, and T(n, 0) = 0^n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 6, 5, 0, 5, 20, 28, 14, 0, 14, 70, 135, 120, 42, 0, 42, 252, 616, 770, 495, 132, 0, 132, 924, 2730, 4368, 4004, 2002, 429, 0, 429, 3432, 11880, 23100, 27300, 19656, 8008, 1430, 0, 1430, 12870, 51051, 116688, 168300, 157080, 92820, 31824, 4862

Offset: 0

Philippe Deléham, Jun 03 2004, Jun 14 2007

			Triangle begins:
  1;
  0,   1;
  0,   1,    2;
  0,   2,    6,     5;
  0,   5,   20,    28,    14;
  0,  14,   70,   135,   120,    42;
  0,  42,  252,   616,   770,   495,   132;
  0, 132,  924,  2730,  4368,  4004,  2002,  429;
  0, 429, 3432, 11880, 23100, 27300, 19656, 8008, 1430; ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
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.
B. Derrida, E. Domany and D. Mukamel, A exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs.(20), (21), p. 672.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Variant of A062991, unsigned and transposed.
See also A234950 for another version.
Columns: A000007 (k=0), 2*A001700 (k=1).
Diagonals: A002694 (k=n-1), A000108 (k=n).
Row sums: A064062 (generalized Catalan C(2; n)).
Cf. A000012, A002694, A064062, A064063, A064087, A064088, A064089.
Cf. A064090, A064091, A064092, A064093, A064094.

A094385:= func< n,k | n eq 0 select 1 else Binomial(2*n, k-1)*Binomial(2*n-k-1, n-k)/n >;
[A094385(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 26 2024

T[n_, k_] := Binomial[2n, k-1] Binomial[2n-k-1, n-k]/n; T[0, 0] = 1;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2018 *)

def A094385(n,k): return 1 if (n==0) else binomial(2*n,k-1)*binomial(2*n-k-1, n-k)//n
flatten([[A094385(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 26 2024

T is given by [0, 1, 1, 1, 1, 1, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.
Sum_{k = 0..n} T(n, k)*x^(n-k) = C(x+1; n), generalized Catalan numbers; see left diagonals of triangle A064094: A000012, A000108, A064062, A064063, A064087..A064093 for x = -1, 0, ..., 9, respectively.
From G. C. Greubel, Sep 26 2024: (Start)
T(n, 1) = A000108(n-1), n >= 1.
T(n, n-1) = A002694(n), n >= 1.
T(n, n) = A000108(n). (End)

New name using a formula of the author by Peter Luschny, Sep 26 2024

A365816 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 3 * A(x)).

Original entry on oeis.org

0, 1, 6, 57, 658, 8442, 115692, 1658505, 24565530, 372999198, 5774883348, 90821581578, 1446901409268, 23301338376916, 378711707274072, 6203898306232233, 102329366764727658, 1698047225583890550, 28327664136201303300, 474821679792884860590, 7992739387298462213340

Offset: 0

Ilya Gutkovskiy, Sep 25 2023

nonn

Reversion of g.f. for hexagonal numbers (with signs).

Eric Weisstein's World of Mathematics, Hexagonal Number
Eric Weisstein's World of Mathematics, Series Reversion

Cf. A000384, A064087, A113207, A179848, A263843, A365817, A365818.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^3/(1 - 3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 3 x)/(1 + x)^3, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[3 n, n - k - 1] 3^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(3*n,n-k-1) * 3^k for n > 0.

a(n) ~ 6^(3*n + 1/2) / (sqrt((481 + 133*sqrt(13))*Pi) * n^(3/2) * (13*sqrt(13) - 35)^n). - Vaclav Kotesovec, Sep 26 2023

A366015 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 3 * A(x)).

Original entry on oeis.org

0, 1, 7, 76, 995, 14433, 223300, 3611016, 60305787, 1032115315, 18007816255, 319110233104, 5727667197044, 103913426353324, 1902498385538520, 35106179258551632, 652236828560562987, 12190651925663309175, 229059610932456616501, 4324334144117016053500, 81983637468108446363755

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Reversion of g.f. for hexagonal pyramidal numbers (with signs).

Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number
Eric Weisstein's World of Mathematics, Series Reversion

Cf. A002293, A002412, A064087, A365754, A365816, A366014, A366016, A366017.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^4/(1 - 3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 3 x)/(1 + x)^4, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[4 n, n - k - 1] 3^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(4*n,n-k-1) * 3^k for n > 0.

A064311 Generalized Catalan numbers C(-2; n).

Original entry on oeis.org

1, 1, -1, 5, -25, 141, -849, 5349, -34825, 232445, -1582081, 10938709, -76616249, 542472685, -3876400305, 27919883205, -202480492905, 1477306676445, -10836099051105, 79861379898165, -591082795606425

Offset: 0

Wolfdieter Lang, Sep 21 2001

See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.

Generalized Catalan numbers C(m; n): A000012 (m = 0), A000108 (m = 1), A064062 (m = 2), A064063 (m = 3), A064087 - A064093 (m = 4 thru 10); A064310 (m = -1) and A064325 - A064333 (m = -3 thru -11).

a[n_] := If[n==0, 1, Sum[(n-m)*Binomial[n+m-1, m]*(-2)^m/n, {m,0,n-1}]];
Table[a[n], {n,0,20}] (* Jean-François Alcover, Jun 03 2019 *)

import mpmath
mp.dps = 25; mp.pretty = True
a = lambda n: mpmath.hyp2f1(1-n, n, -n, -2) if n>0 else 1
[int(a(n)) for n in range(21)] # Peter Luschny, Nov 30 2014

a(n) = (1/n) * Sum_{m = 0..n-1} (n-m)*binomial(n-1+m, m)*(-2)^m = ((1/3)^n)*(1 + 2*Sum_{k = 0..n-1} C(k)*(-2*3)^k), for n >= 1, with a(0) := 1, and where C(n) = A000108(n), the Catalan numbers.
G.f.: (1+2*x*c(-2*x)/3)/(1-x/3) = 1/(1-x*c(-2*x)) with c(x) the g.f. of the Catalan numbers A000108.
a(n) = hypergeom([1-n, n], [-n], -2) for n>0. - Peter Luschny, Nov 30 2014
a(n) ~ -(-1)^n * 2^(3*n+1) / (25 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 03 2019
G.f. A(x) = 1 + series_reversion(x*(1 - (m-1)*x)/(1 + x)^2) at m = -2. - Peter Bala, Sep 08 2024

A323206 A(n, k) = hypergeometric([-k, k+1], [-k-1], n), square array read by ascending antidiagonals for n,k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 13, 14, 1, 1, 5, 25, 67, 42, 1, 1, 6, 41, 190, 381, 132, 1, 1, 7, 61, 413, 1606, 2307, 429, 1, 1, 8, 85, 766, 4641, 14506, 14589, 1430, 1, 1, 9, 113, 1279, 10746, 55797, 137089, 95235, 4862, 1

Offset: 0

Peter Luschny, Feb 21 2019

Conjecture: A(n, k) is odd if and only if n is even or (n is odd and k + 2 = 2^j for some j > 0).

			Array starts:
    [n\k 0  1    2     3       4        5         6           7  ...]
    [0]  1, 1,   1,    1,      1,       1,        1,          1, ... A000012
    [1]  1, 2,   5,   14,     42,     132,      429,       1430, ... A000108
    [2]  1, 3,  13,   67,    381,    2307,    14589,      95235, ... A064062
    [3]  1, 4,  25,  190,   1606,   14506,   137089,    1338790, ... A064063
    [4]  1, 5,  41,  413,   4641,   55797,   702297,    9137549, ... A064087
    [5]  1, 6,  61,  766,  10746,  161376,  2537781,   41260086, ... A064088
    [6]  1, 7,  85, 1279,  21517,  387607,  7312789,  142648495, ... A064089
    [7]  1, 8, 113, 1982,  38886,  817062, 17981769,  409186310, ... A064090
    [8]  1, 9, 145, 2905,  65121, 1563561, 39322929, 1022586105, ... A064091
         A001844 A064096 A064302  A064303   A064304   A064305  diag: A323209
.
Seen as a triangle (by reading ascending antidiagonals):
                               1
                              1, 1
                            1, 2, 1
                           1, 3, 5, 1
                        1, 4, 13, 14, 1
                      1, 5, 25, 67, 42, 1
                   1, 6, 41, 190, 381, 132, 1

J. Abate, W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011).
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.

Rows: A000108, A064062, A064063, A064087, A064088, A064089, A064090, A064091.
Columns: A001844, A064096, A064302, A064303, A064304, A064305.
Diagonals: A323209 (main), A323208 (sup main), A323217 (sub main).
Sums of antidiagonals: A323207
Cf. A064094, A009766, A238762.

# The function ballot is defined in A238762.
A := (n, k) -> add(ballot(2*j, 2*k)*n^j, j=0..k):
for n from 0 to 6 do seq(A(n, k), k=0..9) od;
# Or by recurrence:
A := proc(n, k) option remember;
if n = 1 then return `if`(k = 0, 1, (4*k + 2)*A(1, k-1)/(k + 2)) fi:
if k < 2 then return [1, n+1][k+1] fi; n*(4*k - 2);
((%*(n - 1) - k - 1)*A(n, k-1) + %*A(n, k-2))/((n - 1)*(k + 1)) end:
for n from 0 to 6 do seq(A(n, k), k=0..9) od;
# Alternative:
Arow := proc(n, len) # Function REVERT is in Sloane's 'Transforms'.
[seq(1 + n*k, k=0..len-1)]; REVERT(%); seq((-1)^k*%[k+1], k=0..len-1) end:
for n from 0 to 8 do Arow(n, 8) od;

A[n_, k_] := Hypergeometric2F1[-k, k + 1, -k - 1, n];
Table[A[n, k], {n, 0, 8}, {k, 0, 8}]
(* Alternative: *)
prev[f_, n_] := InverseSeries[Series[-x f, {x, 0, n}]]/(-x);
f[n_, x_] := (1 + (n - 1) x)/((1 - x)^2);
For[n = 0, n < 9, n++, Print[CoefficientList[prev[f[n, x], 8], x]]]
(* Continued fraction: *)
num[k_, n_] := If[k < 2, 1, If[k == 2, -x, -n x]];
cf[n_, len_] := ContinuedFractionK[num[k, n], 1, {k, len + 2}];
Arow[n_, len_] := Rest[CoefficientList[Series[cf[n, len], {x, 0, len}], x]];
For[n = 0, n < 9, n++, Print[Arow[n, 8]]]

{A(n,k) = polcoeff((1/x)*serreverse(x*((1+(n-1)*(-x))/((1-(-x))^2)+x*O(x^k))), k)}
for(n=0, 8, for(k=0, 8, print1(A(n, k), ", ")); print())

# Valid for n > 0.
def genCatalan(n): return SR(1/(x- x^2*(1 - sqrt(1 - 4*x*n))/(2*x*n)))
for n in (1..8): print(genCatalan(n).series(x).list())
# Alternative:
def pseudo_reversion(g, invsign=false):
    if invsign: g = g.subs(x=-x)
    g = g.shift(1)
    g = g.reverse()
    g = g.shift(-1)
    return g
R. = PowerSeriesRing(ZZ)
for n in (0..6):
    f = (1+(n-1)*x)/((1-x)^2)
    s = pseudo_reversion(f, true)
    print(s.list())

A(n, k) = [x^k] 1/(x - x^2*C(n*x)) if n > 0 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the generating function of the Catalan numbers A000108.
A(n, k) = Sum_{j=0..k} (binomial(2*k-j, k) - binomial(2*k-j, k+1))*n^(k-j).
A(n, k) = Sum_{j=0..k} binomial(k + j, k)*(1 - j/(k + 1))*n^j (cf. A009766).
A(n, k) = 1 + Sum_{j=0..k-1} ((1+j)*binomial(2*k-j, k+1)/(k-j))*n^(k-j).
A(n, k) = (1/(2*Pi))*Integral_{x=0..4*n} (sqrt(x*(4*n-x))*x^k)/(1+(n-1)*x), n>0.
A(n, k) ~ ((4*n)^k/(Pi^(1/2)*k^(3/2)))*(1+1/(2*n-1))^2.
If we shift the series f with constant term 1 to the right, invert it with respect to composition and shift the result back to the left then we call this the 'pseudo reversion' of f, prev(f). Row n of the array gives the coefficients of the pseudo reversion of f = (1 + (n - 1)*x)/((1 - x)^2) with an additional inversion of sign. Note that f is not revertible. See also the Sage implementation below.
A(n, k) = [x^k] prev((1 + (n - 1)*(-x))/(1 - (-x))^2).
A(n, k) = [x^(k+1)] cf(n, x) where cf(n, x) = K_{i>=1} c(i)/b(i) in the notation of Gauß with b(i) = 1, c(1) = 1, c(2) = -x and c(i) = -n*x for i > 2.
For a recurrence see the Maple section.

cp's OEIS Frontend

A106566 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, ... ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ... ] where DELTA is the operator defined in A084938.

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A064062 Generalized Catalan numbers C(2; n).

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A064094 Triangle composed of generalized Catalan numbers.

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A064088 Generalized Catalan numbers C(5; n).

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A094385 Triangle read by rows: T(n, k) = binomial(2*n, k-1)*binomial(2*n-k-1, n-k)/n for n, k >= 1, and T(n, 0) = 0^n.

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A365816 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 3 * A(x)).

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A094385 Triangle read by rows: T(n, k) = binomial(2n, k-1)binomial(2*n-k-1, n-k)/n for n, k >= 1, and T(n, 0) = 0^n.