A006318 -id:A006318 - cp's OEIS frontend

A026769 Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; T(2,1)=2; for n >= 3 and 1<=k<=n-1, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if 1<=k<=(n-1)/2, else T(n,k) = T(n-1,k-1) + T(n-1,k).

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 7, 4, 1, 1, 8, 17, 11, 5, 1, 1, 10, 31, 28, 16, 6, 1, 1, 12, 49, 76, 44, 22, 7, 1, 1, 14, 71, 156, 120, 66, 29, 8, 1, 1, 16, 97, 276, 352, 186, 95, 37, 9, 1, 1, 18, 127, 444, 784, 538, 281, 132, 46, 10, 1, 1, 20, 161, 668, 1504, 1674, 819, 413, 178, 56, 11, 1

Offset: 0

Clark Kimberling

T(n, k) is the number of paths from (0, 0) to (k,n-k) in the directed graph having vertices (i, j) (i and j in range [0,n]) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j>=0 and edges (i,i+h)-to-(i+1,i+h+1) for i>=0, h>=1.

Also, square array R read by antidiagonals where R(i,j) = T(i+j,i), which is equal to the number of paths from (0,0) to (i,j) in the above graph. - Max Alekseyev, Dec 02 2015

			Triangle begins as:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   3,   1;
  1,  6,   7,   4,   1;
  1,  8,  17,  11,   5,   1;
  1, 10,  31,  28,  16,   6,   1;
  1, 12,  49,  76,  44,  22,   7,   1;
  1, 14,  71, 156, 120,  66,  29,   8,  1;
  1, 16,  97, 276, 352, 186,  95,  37,  9,  1;
  1, 18, 127, 444, 784, 538, 281, 132, 46, 10, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened
M. A. Alekseyev. On Enumeration of Dyck-Schroeder Paths. Journal of Combinatorial Mathematics and Combinatorial Computing 106 (2018), 59-68; arXiv:1601.06158 [math.CO], 2016-2018.

Cf. A026770, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026779

Cf. A026780 (a variant with h>=0)

T:= function(n,k)
    if k=0 or k=n then return 1;
    elif (n=2 and k=1) then return 2;
    elif (k <= Int((n-1)/2)) then return T(n-1,k-1)+T(n-2,k-1) +T(n-1,k);
    else return T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Oct 31 2019

A026769 := proc(n,k)
    option remember;
    if k= 0 or k =n then
        1;
    elif n= 2 and k= 1 then
        2;
    elif k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    fi ;
end proc: # R. J. Mathar, Jun 15 2014

T[n_, k_] := T[n, k] = Which[k==0 || k==n, 1, n==2 && k==1, 2, k <= (n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], True, T[n-1, k-1] + T[n-1, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 10 2017, from Maple *)

T(n,k) = if(k==0 || k==n, 1, if(n==2 && k==1, 2, if( k<=(n-1)/2, T(n-1,k-1) + T(n-2,k-1) + T(n-1,k), T(n-1,k-1) + T(n-1,k) )));
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 31 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (n==2 and k==1): return 2
    elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 31 2019

For n>=2*k, T(n,k) = coefficient of x^k in G(x)*S(x)^(n-2*k). For n<=2*k, T(n,k) = coefficient of x^(n-k) in G(x)*C(x)^(2*k-n). Here C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108, S(x)=(1-x - sqrt(1-6*x+x^2) )/(2*x) is o.g.f. for A006318, and G(x)=1/(1-x*(C(x)+S(x))) is o.g.f. for A026770. - Max Alekseyev, Dec 02 2015

Offset corrected by R. J. Mathar, Jun 15 2014

More terms added by G. C. Greubel, Oct 31 2019

A073157 Number of Schroeder n-paths containing no FFs.

Original entry on oeis.org

1, 2, 5, 18, 70, 293, 1283, 5808, 26960, 127628, 613814, 2990681, 14730713, 73229291, 366936231, 1851352820, 9397497758, 47957377934, 245903408244, 1266266092112, 6545667052320, 33954266444498, 176689391245146

Offset: 0

Paul D. Hanna, Jul 29 2002

Number of Schroeder n-paths containing no FFs. A Schroeder n-path (A006318) consists of steps U=(1,1),F=(2,0),D=(1,-1) starting at (0,0), ending at (2n,0), and never going below the x-axis. Example: a(2)=5 counts UFD, UUDD, UDF, FUD, UDUD. - David Callan, Aug 23 2011

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 70*x^4 + 293*x^5 + 1283*x^6 + ...

Muniru A Asiru, Table of n, a(n) for n = 0..300

Leftmost column of triangle A073154 (was previous name).

Cf. A000108, A073155, A073156, A073153, A143330, A198953, A215715, A216359, A364335, A364371.

List([0..25],n->Sum([0..n],i->Binomial(2*i+1,i)*Binomial(2*i+1,n-i)/(2*i+1))); # Muniru A Asiru, Oct 11 2018

a:=n->add(binomial(2*i+1,i)*binomial(2*i+1,n-i)/(2*i+1),i=0..n): seq(a(n),n=0..25); # Muniru A Asiru, Oct 11 2018

Table[Sum[Binomial[2*i + 1, i]*Binomial[2*i + 1, n - i]/(2*i + 1), {i, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 11 2018 *)

a(n):=sum((sum((binomial(2*k+2,j-k)*binomial(2*k,k)/(k+1)),k,0,j))*(-1)^(n-j),j,0,n); /* Vladimir Kruchinin, Mar 13 2016 */

{a(n)=local(A=1); for(i=0,n-1,A=(1+x)*(1+x*(A+x*O(x^n))^2));polcoeff(A,n)} /* Paul D. Hanna, Mar 03 2008 */

A073155(n+1) = Sum_{k=0..n} a(k)*a(n-k), that is, convolution yields sequence A073155 minus the 0th term.
G.f.: A(x) = (1 - sqrt(1 - 4*x*(1+x)^2))/(2*x*(1+x)) satisfies A(x) = (1+x)*(1 + x*A(x)^2);
G.f.: A(x) = (1+x)*C(x*(1+x)^2) where C(x) is the Catalan g.f. of A000108. - Paul D. Hanna, Mar 03 2008
a(n) = Sum_{j=0..n}((Sum_{k=0..j}((binomial(2*k+2,j-k)*C(k))))*(-1)^(n-j)), where C(k) = A000108(k). - Vladimir Kruchinin, Mar 13 2016
a(n) = Sum_{i=0..n} C(2*i+1,i)*C(2*i+1,n-i)/(2*i+1). - Vladimir Kruchinin, Oct 11 2018
Recurrence: (n+1)*a(n) = 3*(n-1)*a(n-1) + 6*(2*n - 3)*a(n-2) + 6*(2*n - 5)*a(n-3) + 2*(2*n - 7)*a(n-4). - Vaclav Kotesovec, Oct 11 2018
From Peter Bala, Aug 25 2024: (Start)
(1/x) * series_reversion(x/A(x)) = 1 + 2*x + 9*x^2 + 56*x^3 + 400*x^4 + 3095*x^5 + 25240*x^6 + ... is the g.f. of A198953.
(1/x) * series_reversion(x*A(-x)) = 1 + 2*x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 289*x^6 + ... = G(x) + x, where G(x) = (1 - x^2 - sqrt(1 - 4*x - 2*x^2 + x^4))/(2*x) is the g.f. of A143330. (End)
Define a sequence operator R: {u(n): n >= 0} -> {v(n): n >= 0} by Sum_{n >= 0} v(n)*x^n = (1/x) * series_reversion(x/Sum_{n >= 0} u(n)*x^n). Then R({a(n)}) = A198953, R^2({a(n)}) = A215715 and R^3({a(n)}) = A364335. Cf. A216359. - Peter Bala, Sep 13 2024

More terms from Paul D. Hanna, Mar 03 2008

New name using a comment from David Callan, Peter Luschny, Oct 14 2018

A047891 Number of planar rooted trees with n nodes and tricolored end nodes.

Original entry on oeis.org

1, 3, 12, 57, 300, 1686, 9912, 60213, 374988, 2381322, 15361896, 100389306, 663180024, 4421490924, 29712558576, 201046204173, 1368578002188, 9366084668802, 64403308499592, 444739795023054, 3082969991029800

Offset: 1

Louis Shapiro

Essentially the same as A025231.
Also number of lattice paths from (0,0) to (n-1,n-1), with steps (1,0),(0,1) and (1,1), that never rise above the line y=x and the steps (1,1) are colored red or blue. - Emeric Deutsch, May 28 2003
The Hankel transform (see A001906 for definition) of this sequence forms A049656(n+1) = [1, 3, 27, 729, 59049, 14348907, ...]. - Philippe Deléham, Aug 29 2006
With a(0)=0, this is the series reversion of x(1-x)/(1+2x). - Paul Barry, Oct 18 2009
Row sums of the Riordan matrix A121576. - Emanuele Munarini, May 18 2011

			G.f. = x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1686*x^6 + 9912*x^7 + ...

Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Paul Barry and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
Zhi Chen and Hao Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO] (2016), eq. (1.13), a=3, b=1.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Eric Weisstein's MathWorld, Legendre Polynomial.
Index entries for sequences related to rooted trees

Cf. A006318, A121576, A054872.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-2*x-Sqrt(1-8*x+4*x^2))/(2*x))); // G. C. Greubel, Feb 10 2018

A047891_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 3*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a,list)end: A047891_list(20); # Peter Luschny, May 19 2011

CoefficientList[Series[(1-2x-Sqrt[1-8x+4x^2])/(2x),{x,0,100}],x] (* Emanuele Munarini, May 18 2011 *)
a[ n_] := SeriesCoefficient[(1 - 2 x - Sqrt[1 - 8 x + 4 x^2]) / 2, {x, 0, n}]; (* Michael Somos, Apr 10 2014 *)
Table[2^(n-1) (LegendreP[n, 2] - LegendreP[n-2, 2])/(2n-1), {n, 1, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)
Table[3 Hypergeometric2F1[1-n, 2-n, 2, 3] - 2 KroneckerDelta[n-1], {n, 1, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

makelist(sum(binomial(n,k)*binomial(2*n-k+1,n+1)*(2*n^2-6*(k-1)*n+3*k^2-9*k+4)/((n-k+2)*(n-k+1))*2^k,k,0,n)/2,n,0,24); /* Emanuele Munarini, May 18 2011 */

a(n)=if(n<2,n==1,n--;sum(k=0,n,3^k*binomial(n,k)*binomial(n,k-1))/n)

x='x+O('x^100); Vec((1-2*x-sqrt(1-8*x+4*x^2))/2) \\ Altug Alkan, Nov 02 2015

G.f.: (1 - 2*x - sqrt(1 - 8*x + 4*x^2))/2.
For n>0, a(n+1) = (1/n)*Sum_{k=0..n} 3^k*C(n, k)*C(n, k-1) - Benoit Cloitre, May 10 2003
a(1)=1, a(n) = 2*a(n-1) + Sum_{i=1..(n-1)} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
The Hankel transform (see A001906 for definition) of this sequence form A049656(n+1)= [1, 3, 27, 729, 59049, 14348907, ...]. - Philippe Deléham, Aug 29 2006
2*a(n) = A054872(n+1). - Philippe Deléham, Aug 17 2007
From Paul Barry, Feb 01 2009: (Start)
G.f.: x/(1-2x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-... (continued fraction);
a(n+1) = Sum_{k=0..n} C(n+k,2k)*2^(n-k)*A000108(k). (End)
G.f.: x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-... (continued fraction). - Paul Barry, Oct 18 2009
a(1) = 1, for n>=1, a(n+1) = 3*A007564(n). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009
From Emanuele Munarini, May 18 2011: (Start)
a(n+1) = (Sum_{k=0..n} binomial(n,k)*binomial(2*n-k+1,n+1)*(2*n^2-6*(k-1)*n+3*k^2-9*k+4)/((n-k+2)*(n-k+1))*2^k)/2.
D-finite with recurrence: (n+2)*(n+3)*a(n+3) - 6*(n+2)^2*a(n+2) - 12*(n)^2*a(n+1) + 8*n*(n-1)*a(n) = 0. (End)
G.f.: A(x) = (1-2*x-sqrt(4*x^2-8*x+1))/2 = 1 - G(0); G(k)= 1 + 2*x - 3*x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 05 2012
G.f.: x/W(0), where W(k)= k+1 - 2*x*(k+1) - x*(k+1)*(k+2)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013
From Vladimir Reshetnikov, Nov 01 2015: (Start)
a(n) = 2^(n-1)*(LegendreP_n(2) - LegendreP_{n-2}(2))/(2n-1).
a(n) = 3*hypergeom([1-n,2-n], [2], 3) - 2*0^(n-1). (End)
a(n) = 2^(n-1)*hypergeom([1-n, n], [2], -1/2). - Peter Luschny, Nov 25 2020
a(n) ~ 3^(1/4) * (1 + sqrt(3))^(2*n - 1) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 31 2021
D-finite with recurrence n*a(n) +4*(-2*n+3)*a(n-1) +4*(n-3)*a(n-2)=0. - R. J. Mathar, Aug 01 2022

More terms from Christian G. Bower, Dec 11 1999

A033877 Triangular array read by rows associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 16, 22, 1, 8, 30, 68, 90, 1, 10, 48, 146, 304, 394, 1, 12, 70, 264, 714, 1412, 1806, 1, 14, 96, 430, 1408, 3534, 6752, 8558, 1, 16, 126, 652, 2490, 7432, 17718, 33028, 41586, 1, 18, 160, 938, 4080, 14002, 39152, 89898, 164512, 206098

Offset: 1

N. J. A. Sloane

A106579 is in some ways a better version of this sequence, but since this was entered first it will be the main entry for this triangle.
The diagonals of this triangle are self-convolutions of the main diagonal A006318: 1, 2, 6, 22, 90, 394, 1806, ... . - Philippe Deléham, May 15 2005
From Johannes W. Meijer, Sep 22 2010, Jul 15 2013: (Start)
Note that for the terms T(n,k) of this triangle n indicates the column and k the row.
The triangle sums, see A180662, link Schroeder's triangle with several sequences, see the crossrefs. The mirror of this triangle is A080247.
Quite surprisingly the Kn1p sums, p >= 1, are all related to A026003 and crystal ball sequences for n-dimensional cubic lattices (triangle offset is 0): Kn11(n) = A026003(n), Kn12(n) = A026003(n+2) - 1, Kn13(n) = A026003(n+4) - A005408(n+3), Kn14(n) =  A026003(n+6) - A001844(n+4), Kn15(n) = A026003(n+8) - A001845(n+5), Kn16(n) = A026003(n+10) - A001846(n+6), Kn17(n) = A026003(n+12) - A001847(n+7), Kn18(n) = A026003(n+14) - A001848(n+8), Kn19(n) = A026003(n+16) - A001849(n+9), Kn110(n) = A026003(n+18) - A008417(n+10), Kn111(n) = A026003(n+20) - A008419(n+11), Kn112(n) = A026003(n+22) - A008421(n+12). (End)
T(n,k) is the number of normal semistandard Young tableaux with two columns, one of height k and one of height n. The recursion can be seen by performing jeu de taquin deletion on all instances of the smallest value. (If there are two instances of the smallest value, jeu de taquin deletion will always shorten the right column first and the left column second.) - Jacob Post, Jun 19 2018

			Triangle starts:
  1;
  1,    2;
  1,    4,    6;
  1,    6,   16,   22;
  1,    8,   30,   68,   90;
  1,   10,   48,  146,  304,  394;
  1,   12,   70,  264,  714, 1412, 1806;
  ... - _Joerg Arndt_, Sep 29 2013

T. D. Noe, Rows k = 1..50 of triangle, flattened
Henry Bottomley, Illustration of initial terms
Kevin Brown, Hipparchus on Compound Statements, 1994-2010. - _Johannes W. Meijer_, Sep 22 2010
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
G. Kreweras, Aires des chemins surdiagonaux et application à un problème économique, Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche 24 (1976): 1-8. [Annotated scanned copy]
J. W. Meijer, Famous numbers on a chessboard, Acta Nova, Volume 4, No.4, December 2010. pp. 589-598.
J. M. Oh, An explicit formula for the number of fuzzy subgroups of a finite abelian p-group of rank two, Iranian Journal of Fuzzy Systems, Dec 2013, Vol. 10 Issue 6, pp. 125-135.
E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
S. Samieinia, The number of continuous curves in digital geometry, Port. Math. 67 (1) (2010) 75-89, last table.
R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), Article 03.1.5, 19 pp.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Essentially same triangle as A080247 and A080245 but with rows read in reversed order. Also essentially the same triangle as A106579.
Cf. A000007, A006318, A006319, A006320, A006321, A008288, A080243.
Cf. A084938, A103885, A238112, A330801.
Cf. A001003 (row sums), A026003 (antidiagonal sums).
Triangle sums (see the comments): A001003 (Row1, Row2), A026003 (Kn1p, p >= 1), A006603 (Kn21), A227504 (Kn22), A227505 (Kn23), A006603(2*n) (Kn3), A001850 (Kn4), A227506 (Fi1), A010683 (Fi2).

a033877 n k = a033877_tabl !! n !! k
a033877_row n = a033877_tabl !! n
a033877_tabl = iterate
   (\row -> scanl1 (+) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Apr 17 2013

function t(n,k)
  if k le 0 or k gt n then return 0;
  elif k eq 1 then return 1;
  else return t(n,k-1) + t(n-1,k-1) + t(n-1,k);
  end if;
end function;
[t(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 23 2023

T := proc(n, k) option remember; if n=1 then return(1) fi; if kJohannes W. Meijer, Sep 22 2010, revised Jul 17 2013

Mathematica
```
T[1, ]:= 1; T[n, k_]/;(k
				
```

def A033877_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^k*prec(n, n-k) for k in (0..n-1)]
for n in (1..10): print(A033877_row(n)) # Peter Luschny, Mar 16 2016

@CachedFunction
def t(n, k): # t = A033847
    if (k<0 or k>n): return 0
    elif (k==1): return 1
    else: return t(n, k-1) + t(n-1, k-1) + t(n-1, k)
flatten([[t(n,k) for k in range(1,n+1)] for n in range(1, 16)]) # G. C. Greubel, Mar 23 2023

As an upper right triangle: a(n, k) = a(n, k-1) + a(n-1, k-1) + a(n-1, k) if k >= n >= 0 and a(n, k) = 0 otherwise.
G.f.: Sum T(n, k)*x^n*y^k = (1-x*y-(x^2*y^2-6*x*y+1)^(1/2)) / (x*(2*y+x*y-1+(x^2*y^2-6*x*y+1)^(1/2))). - Vladeta Jovovic, Feb 16 2003
Another version of A000007 DELTA [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] = 1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 22, 0, 1, ..., where DELTA is Deléham's operator defined in A084938.
Sum_{n=1..floor((k+1)/2)} T(n+p-1, k-n+p) = A026003(2*p+k-3) - A008288(2*p+k-3, p-2), p >= 2, k >= 1. - Johannes W. Meijer, Sep 28 2013
From G. C. Greubel, Mar 23 2023: (Start)
(t(n, k) as a lower triangle)
t(n, k) = t(n, k-1) + t(n-1, k-1) + t(n-1, k) with t(n, 1) = 1.
t(n, n) = A006318(n-1).
t(2*n-1, n) = A330801(n-1).
t(2*n-2, n) = A103885(n-1), n > 1.
Sum_{k=1..n-1} t(n, k) = A238112(n), n > 1.
Sum_{k=1..n} t(n, k) = A001003(n).
Sum_{k=1..n-1} (-1)^(k-1)*t(n, k) = (-1)^n*A001003(n-1), n > 1.
Sum_{k=1..n} (-1)^(k-1)*t(n, k) = A080243(n-1).
Sum_{k=1..floor((n+1)/2)} t(n-k+1, k) = A026003(n-1). (End)

More terms from David W. Wilson

A060693 Triangle (0 <= k <= n) read by rows: T(n, k) is the number of Schröder paths from (0,0) to (2n,0) having k peaks.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 10, 6, 1, 14, 35, 30, 10, 1, 42, 126, 140, 70, 15, 1, 132, 462, 630, 420, 140, 21, 1, 429, 1716, 2772, 2310, 1050, 252, 28, 1, 1430, 6435, 12012, 12012, 6930, 2310, 420, 36, 1, 4862, 24310, 51480, 60060, 42042, 18018, 4620, 660, 45, 1, 16796

Offset: 0

F. Chapoton, Apr 20 2001

The rows sum to A006318 (Schroeder numbers), the left column is A000108 (Catalan numbers); the next-to-left column is A001700, the alternating sum in each row but the first is 0.
T(n,k) is the number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k peaks. Example: T(2,1)=3 because we have UU*DD, U*DH and HU*D, the peaks being shown by *. E.g., T(n,k) = binomial(n,k)*binomial(2n-k,n-1)/n for n>0. - Emeric Deutsch, Dec 06 2003
A090181*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 14 2008
T(n,k) is also the number of rooted plane trees with maximal degree 3 and k vertices of degree 2 (a node may have at most 2 children, and there are exactly k nodes with 1 child). Equivalently, T(n,k) is the number of syntactically different expressions that can be formed that use a unary operation k times, a binary operation n-k times, and nothing else (sequence of operands is fixed). - Lars Hellstrom (Lars.Hellstrom(AT)residenset.net), Dec 08 2009

			Triangle begins:
00: [    1]
01: [    1,     1]
02: [    2,     3,      1]
03: [    5,    10,      6,      1]
04: [   14,    35,     30,     10,      1]
05: [   42,   126,    140,     70,     15,      1]
06: [  132,   462,    630,    420,    140,     21,     1]
07: [  429,  1716,   2772,   2310,   1050,    252,    28,    1]
08: [ 1430,  6435,  12012,  12012,   6930,   2310,   420,   36,   1]
09: [ 4862, 24310,  51480,  60060,  42042,  18018,  4620,  660,  45,  1]
10: [16796, 92378, 218790, 291720, 240240, 126126, 42042, 8580, 990, 55, 1]
...

Vincenzo Librandi, Rows n = 0..100, flattened
J. Agapito, A. Mestre, P. Petrullo, and M. Torres, Counting noncrossing partitions via Catalan triangles, CEAFEL Seminar, June 30, 2015.
Jean-Christophe Aval and François Bergeron, Rectangular Schröder Parking Functions Combinatorics, arXiv:1603.09487 [math.CO], 2016.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See p. 25.
David Callan and Toufik Mansour, Five subsets of permutations enumerated as weak sorting permutations, arXiv:1602.05182 [math.CO], 2016.
G. E. Cossali, A Common Generating Function of Catalan Numbers and Other Integer Sequences, J. Int. Seqs. 6 (2003).
Dan Drake, Bijections from Weighted Dyck Paths to Schröder Paths, J. Int. Seq. 13 (2010) # 10.9.2
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
Krishna Menon and Anurag Singh, Grassmannian permutations avoiding identity, arXiv:2212.13794 [math.CO], 2022.
Jean-Christophe Novelli and Jean-Yves Thibon, Duplicial algebras and Lagrange inversion, arXiv preprint arXiv:1209.5959 [math.CO], 2012.

Cf. A006318, A000108, A001700.
Triangle in A088617 transposed.
Diagonals give A000108, A001700, A002457, A002802, A002803; A000012, A000217, A034827, A000910, A088625, A088626.
T(2n,n) gives A007004.
Cf. A001263, A033282, A086810, A088617, A097610, A101282.

A060693 := (n,k) -> binomial(n,k)*binomial(2*n-k,n)/(n-k+1); # Peter Luschny, May 17 2011

t[n_, k_] := Binomial[n, k]*Binomial[2 n - k, n]/(n - k + 1); Flatten[Table[t[n, k], {n, 0, 9}, {k, 0, n}]] (* Robert G. Wilson v, May 30 2011 *)

T(n, k) = binomial(n, k)*binomial(2*n - k, n)/(n - k + 1);
for(n=0, 10, for(k=0, n, print1(T(n, k),", ")); print); \\ Indranil Ghosh, Jul 28 2017

from sympy import binomial
def T(n, k): return binomial(n, k) * binomial(2 * n - k, n) / (n - k + 1)
for n in range(11): print([T(n, k) for k in range(n + 1)])  # Indranil Ghosh, Jul 28 2017

Triangle T(n, k) (0 <= k <= n) read by rows; given by [1, 1, 1, 1, 1, ...] DELTA [1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 12 2003
If C_n(x) is the g.f. of row n of the Narayana numbers (A001263), C_n(x) = Sum_{k=1..n} binomial(n,k-1)*(binomial(n-1,k-1)/k) * x^k and T_n(x) is the g.f. of row n of T(n,k), then T_n(x) = C_n(x+1), or T(n,k) = [x^n]Sum_{k=1..n}(A001263(n,k)*(x+1)^k). - Mitch Harris, Jan 16 2007, Jan 31 2007
G.f.: (1 - t*y - sqrt((1-y*t)^2 - 4*y)) / 2.
T(n, k) = binomial(2n-k, n)*binomial(n, k)/(n-k+1). - Philippe Deléham, Dec 07 2003
A060693(n, k) = binomial(2*n-k, k)*A000108(n-k); A000108: Catalan numbers. - Philippe Deléham, Dec 30 2003
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n), for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Apr 01 2007
T(n,k) = Sum_{j>=0} A090181(n,j)*binomial(j,k). - Philippe Deléham, May 04 2007
Sum_{k=0..n} T(n,k)*x^(n-k) = (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Oct 18 2007
From Paul Barry, Jan 29 2009: (Start)
G.f.: 1/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-.... (continued fraction);
G.f.: 1/(1-(x+xy)/(1-x/(1-(x+xy)/(1-x/(1-(x+xy)/(1-.... (continued fraction). (End)
T(n,k) = [k<=n]*(Sum_{j=0..n} binomial(n,j)^2*binomial(j,k))/(n-k+1). - Paul Barry, May 28 2009
T(n,k) = A104684(n,k)/(n-k+1). - Peter Luschny, May 17 2011
From Tom Copeland, Sep 21 2011: (Start)
With F(x,t) = (1-(2+t)*x-sqrt(1-2*(2+t)*x+(t*x)^2))/(2*x) an o.g.f. (nulling the n=0 term) in x for the A060693 polynomials in t,
  G(x,t) = x/(1+t+(2+t)*x+x^2) is the compositional inverse in x.
Consequently, with H(x,t) = 1/(dG(x,t)/dx) = (1+t+(2+t)*x+x^2)^2 / (1+t-x^2), the n-th A060693 polynomial in t is given by (1/n!)*((H(x,t)*d/dx)^n) x evaluated at x=0, i.e., F(x,t) = exp(x*H(u,t)*d/d) u, evaluated at u = 0.
  Also, dF(x,t)/dx = H(F(x,t),t). (End)
See my 2008 formulas in A033282 to relate this entry to A088617, A001263, A086810, and other matrices. - Tom Copeland, Jan 22 2016
Rows of this entry are non-vanishing antidiagonals of A097610. See p. 14 of Agapito et al. for a bivariate generating function and its inverse. - Tom Copeland, Feb 03 2016
From Werner Schulte, Jan 09 2017: (Start)
T(n,k) = A126216(n,k-1) + A126216(n,k) for 0 < k < n;
Sum_{k=0..n} (-1)^k*(1+x*(n-k))*T(n,k) = x + (1-x)*A000007(n).
(End)
Conjecture: Sum_{k=0..n} (-1)^k*T(n,k)*(n+1-k)^2 = 1+n+n^2. - Werner Schulte, Jan 11 2017

More terms from Vladeta Jovovic, Apr 21 2001
New description from Philippe Deléham, Aug 12 2003
New name using a comment by Emeric Deutsch from Peter Luschny, Jul 26 2017

A346626 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^3) / (1 - x).

Original entry on oeis.org

1, 2, 8, 44, 280, 1936, 14128, 107088, 834912, 6652608, 53934080, 443467136, 3689334272, 30997608960, 262651640064, 2241857334528, 19257951946240, 166362924583936, 1444351689281536, 12595885932259328, 110287974501355520, 969178569410404352, 8544982917273509888, 75565732555028701184

Offset: 0

Ilya Gutkovskiy, Jul 25 2021

nonn

Partial sums of A213282.

Seiichi Manyama, Table of n, a(n) for n = 0..500
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See p. 26.

Cf. A001764, A006318, A199475, A213282, A349310, A349311, A349312, A349313, A349314.

nmax = 23; A[] = 0; Do[A[x] = (1 + x A[x]^3)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 23; CoefficientList[Series[Sum[(Binomial[3 k, k]/(2 k + 1)) x^k/(1 - x)^(3 k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 23}]

G.f.: Sum_{k>=0} ( binomial(3*k,k) / (2*k + 1) ) * x^k / (1 - x)^(3*k+1).
a(0) = 1; a(n) = a(n-1) + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) ~ 2^(n - 1/2) / (sqrt(3*Pi*(2 - (2 - sqrt(2))^(1/3)/2^(2/3) - 1/(2*(2 - sqrt(2)))^(1/3))) * n^(3/2) * (2 - 3/(sqrt(2) - 1)^(1/3) + 3*(sqrt(2) - 1)^(1/3))^n). - Vaclav Kotesovec, Nov 04 2021
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 2^(n-k) * binomial(n,k) * binomial(2*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024

A002003 a(n) = 2 * Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k).

Original entry on oeis.org

0, 2, 8, 38, 192, 1002, 5336, 28814, 157184, 864146, 4780008, 26572086, 148321344, 830764794, 4666890936, 26283115038, 148348809216, 838944980514, 4752575891144, 26964373486406, 153196621856192, 871460014012682, 4962895187697048, 28292329581548718

Offset: 0

N. J. A. Sloane

nonn

a(n) is the number of order-preserving partial self maps of {1,...,n}. For example, a(2) = 8 because there are 8 order-preserving partial self maps of {1,2}: (1 2), (1 1), (2 2), (1 -), (2 -), (- 1), (- 2), (- -). Here for example (2 -) represents the partial map which maps 1 to 2 but does not include 2 in its domain. - James East, Oct 25 2005
From Peter Bala, Mar 02 2020: (Start)
For fixed m = 1,2,3,..., we conjecture that the sequence b(n) := a(m*n) satisfies a recurrence of the form P(2*m,n)*b(n+1) + P(2*m,-n)*b(n-1) = Q(2*m,n)*b(n), where the polynomials P(2*m,n) and Q(2*m,n) have degree 2*m. Conjecturally, the polynomial Q(2*m,n) is an even function of n; its 2*m zeros seem to belong to the interval [-1, 1] and 2*m - 2 of these zeros appear to lie close to the rational numbers of the form +-(2*k + 1)/(2*m), where 0 <= k <= m - 2. Cf. A103885. (End)
a(n), n>0, is the number of points at L1 distance = n from any given point in Z^n.  The sequence is also the difference between the central diagonal (A001850) and +-1 diagonal (A002002) of the Delannoy number triangle (A008288).  - Shel Kaphan, Feb 15 2023

			G.f. = 2*x + 8*x^2 + 38*x^3 + 192*x^4 + 1002*x^5 + 5336*x^6 + 28814*x^7 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
Peter Bala, A supercongruence for A002003
J. Brzozowski, M. Szykula, Large Aperiodic Semigroups, arXiv preprint arXiv:1401.0157 [cs.FL], 2013-2014.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
G. Rutledge and R. D. Douglass, Integral functions associated with certain binomial coefficient sums, Amer. Math. Monthly, 43 (1936), 27-32.

Cf. A002002, A006318, A027307, A108424, A103885.

A064861 := proc(n,k) option remember; if n = 1 then 1; elif k = 0 then 0; else A064861(n,k-1)+(3/2-1/2*(-1)^(n+k))*A064861(n-1,k); fi; end; seq(A064861(i,i-1),i=1..40);

Flatten[{0,Table[SeriesCoefficient[((1+x)/Sqrt[1-6*x+x^2]-1)/2,{x,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 04 2012 *)
a[ n_] := If[ n < 1, 0, Hypergeometric2F1[ n, -n, 1, -1]]; (* Michael Somos, Aug 24 2014 *)
Table[2*Sum[Binomial[n-1,k]Binomial[n+k,k],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Sep 18 2024 *)

{a(n) = if( n<1, 0, polcoeff( ((1 - x^2) / (1 - x)^2 + x * O(x^n))^n, n))} /* Michael Somos, Sep 24 2003 */

from math import comb
def A002003(n): return sum(comb(n,k)**2*k<Chai Wah Wu, Mar 22 2023

a(n) = 2*A047781(n).
From Vladeta Jovovic, Mar 28 2004: (Start)
G.f.: ((1+x)/sqrt(1-6*x+x^2)-1)/2.
E.g.f.: exp(3*x)*(2*BesselI(0, 2*sqrt(2)*x)+sqrt(2)*BesselI(1, 2*sqrt(2)*x)). (End)
a(n) = T(n, n-1), array T as in A064861.
a(n) = T(n, n-2), array T as in A049600.
a(n+1) = A110110(2n+1). - Tilman Neumann, Feb 05 2009
a(n) = 2 * JacobiP(n-1,0,1,3) = ((7*n+3)*LegendreP(n,3) - (n+1)*LegendreP(n+1,3)) /(2*n) for n > 0. - Mark van Hoeij, Jul 12 2010
Logarithmic derivative of A006318, the large Schroeder numbers. - Paul D. Hanna, Oct 25 2010
D-finite with recurrence: 4*(3*n^2-6*n+2)*a(n-1) - (n-2)*(2*n-1)*a(n-2) - n*(2*n-3)*a(n)=0. - Vaclav Kotesovec, Oct 04 2012
a(n) ~ (3+2*sqrt(2))^n/(2^(3/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 04 2012
Recurrence (an alternative): n*a(n) = (6-n)*a(n-6) + 2*(5*n-27)*a(n-5) + (84-15*n)*a(n-4) + 52*(3-n)*a(n-3) + 3*(2-5*n)*a(n-2) + 2*(5*n-3)*a(n-1), n>=7. - Fung Lam, Feb 05 2014
a(n) = Hyper2F1([-n, n], [1], -1) for n > 0. - Peter Luschny, Aug 02 2014
a(n) = [x^n] ((1+x)/(1-x))^n for n > 0. - Seiichi Manyama, Jun 07 2018
From Peter Bala, Mar 13 2020: (Start)
a(n) = 2 * Sum_{k = 0..n-1}  2^k*C(n,k+1)*C(n-1,k).
a(n) = 2 * (-1)^(n+1) * Sum_{k = 0..n-1} (-2)^k*C(n+k,n-1)*C(n-1,k).
a(n) = Sum_{k = 0..n} C(n,k)*C(2*n-k-1,n-1).
Conjecture: a(n) = - [x^n] (1 - F(x))^n, where F(x) = 2*x + 6*x^2 + 34*x^3 + 238*x^4 + ... is the o.g.f. of A108424. Equivalently, a(n) = -[x^n](G(x))^(-n), where G(x) = 1 + 2*x + 10*x^2 + 66*x^3 + 498*x^4 + ... is the o.g.f. of A027307.
a(p) == 2 ( mod p^3 ) for prime p >= 5. (End)
a(n) = Sum_{k = 1..n} C(n, k) * C(n-1, k-1) * 2^k. - Michael Somos, May 23 2021
a(n) = A001850(n) - A002002(n), for n > 0. - Shel Kaphan, Feb 15 2023

More terms from Barbara Haas Margolius (b.margolius(AT)csuohio.edu), Oct 10 2001

A103885 a(n) = [x^(2*n)] ((1 + x)/(1 - x))^n.

Original entry on oeis.org

1, 2, 16, 146, 1408, 14002, 142000, 1459810, 15158272, 158611106, 1669752016, 17664712562, 187641279616, 2000029880786, 21380213588848, 229129634462146, 2460955893981184, 26482855453375042, 285475524009208720, 3082024598888203090, 33319523640218177408

Offset: 0

Ralf Stephan, Feb 20 2005

From Peter Bala, Mar 01 2020: (Start)
The recurrence given below can be rewritten in the form
(2*n+1)*(2*n+2)*P(2,n)*a(n+1) - (2*n-1)*(2*n-2)*P(2,-n)*a(n-1) = Q(2,n^2)*a(n), where the polynomial Q(2,n) = 4*(55*n^2 - 34*n + 3) and the polynomial P(2,n) = 5*n^2 - 5*n + 1 satisfies the symmetry condition P(2,n) = P(2,1-n) and has real zeros.
More generally, for fixed m = 1,2,3,..., we conjecture that the sequence b(n) := a(m*n) satisfies a recurrence of the form ( Product_{k = 1..2*m} (2*m*n + k) ) * P(2*m,n)*b(n+1) + (-1)^m*( Product_{k = 1..2*m} (2*m*n - k) ) * P(2*m,-n)*b(n-1) = Q(2*m,n^2)*b(n), where the polynomials P(2*m,n) and Q(2*m,n) have degree 2*m. Conjecturally, the polynomial P(2*m,n) = P(2*m,1-n) and has real zeros in the interval [0, 1]. The 4*m zeros of the polynomial Q(2*m,n^2) seem to belong to the interval [-1, 1] and 4*m - 2 of these zeros appear to be approximated by the rational numbers +- k/(3*m), where 1 <= k <= 3*m - 2, k not a multiple of 3. (End)

G. C. Greubel, Table of n, a(n) for n = 0..950 [a(0) = 1 inserted by _Georg Fischer_, Apr 03 2020]
Peter Bala, Notes on A103885
V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

Cf. A002003, A006318, A027307, A103882, A103884, A123164, A144097, A156894, A266213, A370102.

A103885:= func< n | n eq 0 select 1 else (&+[ Binomial(n, k)*Binomial(2*n+k-1, n-1): k in [0..n]]) >;
[A103885(n): n in [0..40]]; // G. C. Greubel, Oct 27 2024

a := n -> `if`(n=0, 1, 2*n*hypergeom([1 - 2*n, 1 - n], [2], 2)):
seq(simplify(a(n)), n=0..17); # Peter Luschny, Dec 30 2019
# Alternative (after Peter Bala ):
gf := n -> ( (1 + x)/(1 - x) )^n: ser := n -> series(gf(n), x, 40):
seq(coeff(ser(n), x, 2*n), n=0..17); # Peter Luschny, Mar 20 2020

Prepend[Table[Sum[2^i Binomial[n, i] Binomial[2n-1, i-1], {i, 1, 2n}], {n,1,20}], 1] (* Vaclav Kotesovec, Jul 01 2015 *)

a(n) = if (n==0, 1, sum(i=0, n, 2^i * binomial(n, i) * binomial(2*n-1, i-1))); \\ Michel Marcus, Mar 21 2020

def A103885(n): return 1 if n==0 else sum(binomial(n, k)*binomial(2*n+k-1, n-1) for k in range(n+1))
[A103885(n) for n in range(41)] # G. C. Greubel, Oct 27 2024

a(n) = Sum_{i=0..n} 2^i * binomial(n,i) * binomial(2*n-1,i-1). [Original definition, with summation range {i=1..n}.]
a(n) = A103884(n, n).
G.f.: A(x) = x*B(x)'/B(x), where B(x) is g.f. of A027307. - Vladimir Kruchinin, Jun 30 2015
From Vaclav Kotesovec, Jul 01 2015: (Start)
Recurrence: n*(2*n-1)*(5*n^2 - 15*n + 11)*a(n) = 2*(55*n^4 - 220*n^3 + 296*n^2 - 152*n + 24)*a(n-1) + (n-2)*(2*n-3)*(5*n^2 - 5*n + 1)*a(n-2).
a(n) ~ ((11 + 5*sqrt(5))/2)^n / (2 * 5^(1/4) * sqrt(Pi*n)). (End)
a(n) = [x^n] (1/(1 - x - x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - ...))))))^n, a continued fraction. - Ilya Gutkovskiy, Sep 29 2017
a(n) = 2*n*hypergeom([1 - 2*n, 1 - n], [2], 2) for n >= 1. - Peter Luschny, Dec 30 2019
From Peter Bala, Mar 01 2020: (Start)
a(n) = Sum_{k = 0..n} C(n, k)*C(2*n+k-1, n-1), with a(0) = 1.
a(n) = Sum_{k = 0..n} C(2*n, 2*k)*C(2*n-k-1, n-1), with a(0) = 1.
a(n) = (1/2)*Sum_{k = 0..n} C(2*n, n-k)*C(2*n+k-1, k). Cf. A156894.
a(n) = [x^n] S(x)^n, where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the sequence of large Schröder numbers A006318.
a(n) = (1/2) * [x^(n)] ( (1 + x)/(1 - x) )^(2*n). Cf. A002003(n) = [x^n] ( (1 + x)/(1 - x) )^n.
Conjecture: a(n) = - [x^n] G(x)^(-n), where G(x) = 1 + 2*x + 14*x^2 + 134*x^3 + 1482*x^4 + ... is the o.g.f. of A144097.
a(p) == 2 ( mod p^3 ) for prime p >= 5. (End)
From Peter Bala, Sep 22 2021: (Start)
a(n) = Sum_{k = 0..n} 4^k*binomial(n+k-1,n)*binomial(n,k)^2 / binomial(2*k,k).
Equivalently, a(n) = [x^n] T(n,(1+x)/(1-x)), where T(n,x) is the n-th Chebyshev polynomial of the first kind. Cf. A103882. (End)
For n>0, a(n) = (1/3) * [x^n] (1/S(-x))^(3*n), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the sequence of large Schröder numbers A006318. Cf. A370102. - Peter Bala, Jul 29 2024

a(0) = 1 added and new name by Peter Bala, Mar 01 2020

A144097 The 4-Schroeder numbers: a(n) = number of lattice paths (Schroeder paths) from (0,0) to (3n,n) with unit steps N=(0,1), E=(1,0) and D=(1,1) staying weakly above the line y = 3x.

Original entry on oeis.org

1, 2, 14, 134, 1482, 17818, 226214, 2984206, 40503890, 561957362, 7934063678, 113622696470, 1646501710362, 24098174350986, 355715715691350, 5289547733908510, 79163575684710818, 1191491384838325474

Offset: 0

Joachim Schroeder (schroderjd(AT)qwa.uovs.ac.za), Sep 10 2008

a(n) is also the number of lattice path from (0,0) to (4n,0) with unit steps (1,3), (2,2) and (1,-1) staying weakly above the x-axis.
Also, the number of planar rooted trees with n non-leaf vertices such that each non-leaf vertex has either 3 or 4 children. - Cameron Marcott, Sep 18 2013
a(n) equals the number of ordered complete 4-ary trees with 3*n + 1 leaves, where the internal vertices come in two colors and such that each vertex and its rightmost child have different colors. See Drake, Example 1.6.9. - Peter Bala, Apr 30 2023

			a(2)=14, because
  01: NNNENNNE,
  02: NNDNNNE,
  03: NNNENND,
  04: NNDNND,
  05: NNNDNNE,
  06: NNNDND,
  07: NNNNENNE,
  08: NNNNEND,
  09: NNNNDNE,
  10: NNNNDD,
  11: NNNNNENE,
  12: NNNNNED,
  13: NNNNNDE,
  14: NNNNNNEE
are all the paths from (0,0) to (2,6) with steps N,E and D weakly above y=3x.

Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.

Alois P. Heinz, Table of n, a(n) for n = 0..800
D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510.08036 [math.CO], 2015.
B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.6.9), A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Available online 30 March 2015.
J. Schroeder, Generalized Schroeder numbers and the rotation principle, Journal of Integer Sequences 10(7).
R. A. Sulanke, A recurrence restricted by a diagonal condition: generalized Catalan arrays, Fibonacci Q., 27 (1989), 33-46.
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 11.
Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin; improperly cited as A144079)

Cf. A027307 (the case y=2x), A008288 (Delannoy numbers), A008412 (4-dimensional coordination numbers).
This appears to equal 2*A243675. - N. J. A. Sloane, Mar 28 2021
The sequences listed in Yang-Jiang's Table 1 appear to be A006318, A001003, A027307, A034015, A144097, A243675, A260332, A243676. - N. J. A. Sloane, Mar 28 2021

Schr:=proc(n,m,l)(n-l*m+1)/m*sum(2^v*binomial(m,v)*binomial(n,v-1),v=1..m) end proc; where n=3m and l=3, also
Schr:=proc(n,m,l)(n-l*m+1)/(n+1)*sum(2^v*binomial(m-1,v-1)*binomial(n+1,v),v=0..m) end proc; where n=3m and l=3, also
Schr:=proc(n,m,l)(n-l*m+1)/m*sum(binomial(m,v)*binomial(n+v,m-1),v=0..m) end proc; where n=3m and l=3, also
Schr:=proc(n,m,l)(n-l*m+1)/(n+1)*sum(binomial(n+1,v)*binomial(m-1+v,n),v=0..n+1) end proc; where n=3m and l=3.
# alternative Maple program:
a:= proc(n) option remember; `if`(n<2, n+1,
      ((15610*n^5 -67123*n^4 +106824*n^3 -77633*n^2
       +25514*n-3000)*a(n-1) -(3*(n-2))*(3*n-4)*
       (3*n-5)*(35*n^2-28*n+5)*a(n-2)) / ((3*(3*n-1))
       *(3*n+1)*n*(35*n^2-98*n+68)))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, May 26 2015

d[n_, k_] := Binomial[n+k, k] Hypergeometric2F1[-k, -n, -n-k, -1]; a[0] = 1; a[n_] = d[3n, n] - 3d[3n+1, n-1] - 2d[3n, n-1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 25 2017 *)

{a(n) = sum(k=0, n, binomial(n, k) * binomial(3*n+k+1, n)/(3*n+k+1))} \\ Seiichi Manyama, Jul 25 2020

{a(n) = if(n==0, 1, sum(k=1, n, 2^k*binomial(n, k) * binomial(3*n, k-1)/n))} \\ Seiichi Manyama, Jul 25 2020

G.f. A(z) satisfies A(z) = 1 + z(A(z)^3 + A(z)^4) a(n)= S_{3n+1}(n) - 3S_n(3n + 1), where S_a(b) are coordination numbers, i.e., the number of points in the a-dimensional cubic lattice Z^a having distance b in the L_1 norm.
Also a(n) = D(3n,n) - 3D(3n + 1,n-1) - 2D(3n,n-1), where D(a,b) are the Delannoy numbers, i.e., the number of paths with N, E and D steps from (0,0) to (a,b).
D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*(35*n^2-98*n+68) *a(n) +(-15610*n^5+67123*n^4-106824*n^3+77633*n^2-25514*n+3000)*a(n-1) +3*(n-2) *(3*n-4) *(3*n-5) *(35*n^2-28*n+5) *a(n-2)=0. - R. J. Mathar, Sep 06 2016
From Seiichi Manyama, Jul 25 2020: (Start)
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*n+k+1, n)/(3*n+k+1).
a(n) = (1/n) * Sum_{k=1..n} 2^k * binomial(n,k) * binomial(3*n,k-1) for n > 0. (End)
a(n) ~ sqrt(12160 + 3853*sqrt(10)) * 3^(3*n - 9/2) / (2*sqrt(5*Pi) * n^(3/2) * (223 - 70*sqrt(10))^(n - 1/2)). - Vaclav Kotesovec, Jul 31 2021
Series reversion of x*(1 - x^3)/(1 + x^3) = x + 2*x^4 + 14*x^7 + 134*x^10 + ... = Sum_{n >= 0} a(n)*x^(3*n+1). - Peter Bala, Apr 30 2023
From Peter Bala, Jun 16 2023: (Start)
The g.f. A(x) = 1 + 2*x + 14*x^2 + 134*x^3 + ... satisfies A(x)^3 = (1/x) * the series reversion of ((1 - x)/(1 + x))^3.
Define b(n) = [x^(3*n)] ( (1 + x)/(1 - x) )^n = (1/3) * [x^n] ((1 + x)/(1 - x))^(3*n) = A333715(n). Then A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ).
a(n) = 2*hypergeom([1 - n, -3*n], [2], 2) for n >= 1. (End)
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 09 2023

A260332 Labelings of n diamond-shaped posets with 4 vertices per diamond where the labels follow the poset relations whose associated reading permutation avoids 231 in the classical sense.

Original entry on oeis.org

1, 2, 18, 226, 3298, 52450, 881970

Offset: 0

Manda Riehl, Jul 29 2015

According to Yang-Jiang (2021) these are the 5-Schroeder numbers. If confirmed, this will prove Michael Weiner's conjectures and enable us to extend the sequence. Yang & Jiang (2021) give an explicit formula for the m-Schroeder numbers in Theorem 2.4. - N. J. A. Sloane, Mar 28 2021
By diamond-shaped poset with 4 vertices, we mean a poset on four elements with e_1 < e_2, e_1 < e_3, e_2 < e_4, e_3 < e_4, and no order relations between e_2 and e_3. In the Hasse diagram for such a poset, we have a least element, two elements in the level above, and one element in the top level, so the diagram resembles a diamond.  The associated permutation is the permutation obtained by reading the labels of each poset by levels left to right, starting with the least element.
Also the number of labelings of n diamond-shaped posets with 4 vertices per diamond where the labels follow the poset relations whose associated reading permutation avoids 312 in the classical sense via reverse complement Wilf equivalence.
Conjecture: Also the number of lattice paths (Schroeder paths) from (0,0) to (n,4n) with unit steps N=(0,1), E=(1,0) and D=(1,1) staying weakly above the line y = 4x. - Michael D. Weiner, Jul 24 2019

			For a single diamond (n=1) poset with 4 vertices, we must label the least element 1 and the greatest element 4, and the two central elements can be labeled either 2, 3 or 3, 2 respectively. Thus the associated permutations are 1234 and 1324.

Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.

M. Paukner, L. Pepin, M. Riehl, and J. Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015.
Manda Riehl, A 231-avoiding diamond whose associated permutation is 1234.
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 11.
Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)

Cf. A260331, A260579.

The sequences listed in Yang-Jiang's Table 1 appear to be A006318, A001003, A027307, A034015, A144097, A243675, A260332, A243676. - N. J. A. Sloane, Mar 28 2021

There is a complicated recursive formula available in Paukner et al.
Yang & Jiang (2021) give an explicit formula for the 5-Schroeder numbers in Theorem 2.4. - N. J. A. Sloane, Mar 28 2021
Conjecture: a(n) = Sum_{k=1..n} binomial(n,k)*binomial(4*n,k-1)*2^k/n for n > 0. - Michael D. Weiner, Jul 23 2019
From Peter Bala, Jun 16 2023: (Start)
Conjectures: 1) the g.f. A(x) = 1 + 2*x + 18*x^2 + 226*x^3 + ... satisfies A(x)^4 = (1/x) * the series reversion of ((1 - x)/(1 + x))^4.
2) Define b(n) = (1/4) * [x^n] ((1 + x)/(1 - x))^(4*n). Then A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ).
3) a(n) = 2 * hypergeom([1 - n, -4*n], [2], 2) for n >= 1 (equivalent to Weiner's conjecture above).
4) [x^n] A(x)^n =  (2*n) * hypergeom([1 - n, 1 - 5*n], [2], 2) for n >= 1. (End)

cp's OEIS Frontend

A026769 Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; T(2,1)=2; for n >= 3 and 1<=k<=n-1, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if 1<=k<=(n-1)/2, else T(n,k) = T(n-1,k-1) + T(n-1,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Sage

Formula

Extensions

A073157 Number of Schroeder n-paths containing no FFs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

Maxima

PARI

Formula

Extensions

A047891 Number of planar rooted trees with n nodes and tricolored end nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

PARI

Formula

Extensions

A033877 Triangular array read by rows associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Sage

SageMath

Formula

Extensions

A060693 Triangle (0 <= k <= n) read by rows: T(n, k) is the number of Schröder paths from (0,0) to (2n,0) having k peaks.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI