A129123 -id:A129123 - cp's OEIS frontend

A053121 Catalan triangle (with 0's) read by rows.

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 3, 0, 1, 0, 5, 0, 4, 0, 1, 5, 0, 9, 0, 5, 0, 1, 0, 14, 0, 14, 0, 6, 0, 1, 14, 0, 28, 0, 20, 0, 7, 0, 1, 0, 42, 0, 48, 0, 27, 0, 8, 0, 1, 42, 0, 90, 0, 75, 0, 35, 0, 9, 0, 1, 0, 132, 0, 165, 0, 110, 0, 44, 0, 10, 0, 1, 132, 0, 297, 0, 275, 0, 154, 0, 54, 0, 11, 0

Offset: 0

Wolfdieter Lang

Inverse lower triangular matrix of A049310(n,m) (coefficients of Chebyshev's S polynomials).
Walks with a wall: triangle of number of n-step walks from (0,0) to (n,m) where each step goes from (a,b) to (a+1,b+1) or (a+1,b-1) and the path stays in the nonnegative quadrant.
T(n,m) is the number of left factors of Dyck paths of length n ending at height m. Example: T(4,2)=3 because we have UDUU, UUDU, and UUUD, where U=(1,1) and D=(1,-1). (This is basically a different formulation of the previous - walks with a wall - property.) - Emeric Deutsch, Jun 16 2011
"The Catalan triangle is formed in the same manner as Pascal's triangle, except that no number may appear on the left of the vertical bar." [Conway and Smith]
G.f. for row polynomials p(n,x) := Sum_{m=0..n} (a(n,m)*x^m): c(z^2)/(1-x*z*c(z^2)). Row sums (x=1): A001405 (central binomial).
In the language of the Shapiro et al. reference such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. The g.f. Ginv(x) of the m=0 column of the inverse of a given Bell-matrix (here A049310) is obtained from its g.f. of the m=0 column (here G(x)=1/(1+x^2)) by Ginv(x)=(f^{(-1)}(x))/x, with f(x) := x*G(x) and f^{(-1)}is the compositional inverse function of f (here one finds, with Ginv(0)=1, c(x^2)). See the Shapiro et al. reference.
Number of involutions of {1,2,...,n} that avoid the patterns 132 and have exactly k fixed points. Example: T(4,2)=3 because we have 2134, 4231 and 3214. Number of involutions of {1,2,...,n} that avoid the patterns 321 and have exactly k fixed points. Example: T(4,2)=3 because we have 1243, 1324 and 2134. Number of involutions of {1,2,...,n} that avoid the patterns 213 and have exactly k fixed points. Example: T(4,2)=3 because we have 1243, 1432 and 4231. - Emeric Deutsch, Oct 12 2006
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0) -> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
Riordan array (c(x^2),xc(x^2)), where c(x) is the g.f. of Catalan numbers A000108. - Philippe Deléham, Nov 25 2007
A053121^2 = triangle A145973. Convolved with A001405 = triangle A153585. - Gary W. Adamson, Dec 28 2008
By columns without the zeros, n-th row = A000108 convolved with itself n times; equivalent to A = (1 + x + 2x^2 + 5x^3 + 14x^4 + ...), then n-th row = coefficients of A^(n+1). - Gary W. Adamson, May 13 2009
Triangle read by rows,product of A130595 and A064189 considered as infinite lower triangular arrays; A053121 = A130595*A064189 = B^(-1)*A097609*B where B = A007318. - Philippe Deléham, Dec 07 2009
From Mark Dols, Aug 17 2010: (Start)
As an upper right triangle, rows represent powers of 5-sqrt(24):
  5 - sqrt(24)^1 = 0.101020514...
  5 - sqrt(24)^2 = 0.010205144...
  5 - sqrt(24)^3 = 0.001030928...
(Divided by sqrt(96) these powers give a decimal representation of the columns of A007318, with 1/sqrt(96) being the middle column.) (End)
T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) having k (1,0)-steps. Example: T(5,3)=4 because, denoting U=(1,1), D=(1,-1), H=(1,0), we have HHHUD, HHUDH, HUDHH, and UDHHH. - Emeric Deutsch, Jun 01 2011
Let S(N,x) denote the N-th Chebyshev S-polynomial in x (see A049310, cf. [W. Lang]). Then x^n = sum_{k=0..n} T(n,k)*S(k,x). - L. Edson Jeffery, Sep 06 2012
This triangle a(n,m) appears also in the (unreduced) formula for the powers rho(N)^n for the algebraic number over the rationals rho(N) = 2*cos(Pi/N) = R(N, 2), the smallest diagonal/side ratio R in the regular N-gon:
  rho(N)^n = sum(a(n,m)*R(N,m+1),m=0..n), n>=0, identical in N >= 1. R(N,j) = S(j-1, x=rho(N)) (Chebyshev S (A049310)). See a comment on this under A039599 (even powers) and A039598 (odd powers). Proof: see the Sep 06 2012 comment by L. Edson Jeffery, which follows from T(n,k) (called here a(n,k)) being the inverse of the Riordan triangle A049310. - Wolfdieter Lang, Sep 21 2013
The so-called A-sequence for this Riordan triangle of the Bell type (c(x^2), x*c(x^2)) (see comments above) is A(x) = 1 + x^2. This proves the recurrence given in the formula section by Henry Bottomley for a(n, m) = a(n-1, m-1) + a(n-1, m+1) for n>=1 and m>=1, with inputs. The Z-sequence for this Riordan triangle is Z(x) = x which proves the recurrence a(n,0) = a(n-1,1), n>=1, a(0,0) = 1. For A- and Z-sequences for Riordan triangles see the W. Lang link under A006232. - Wolfdieter Lang, Sep 22 2013
Rows of the triangle describe decompositions of tensor powers of the standard (2-dimensional) representation of the Lie algebra sl(2) into irreducibles. Thus a(n,m) is the multiplicity of the m-th ((m+1)-dimensional) irreducible representation in the n-th tensor power of the standard one. - Mamuka Jibladze, May 26 2015
The Riordan row polynomials p(n, x) belong to the Boas-Buck class (see a comment and references in A046521), hence they satisfy the Boas-Buck identity: (E_x - n*1)*p(n, x) = (E_x + 1)*Sum_{j=0..n-1} (1/2)*(1 - (-1)^j)*binomial(j+1, (j+1)/2)*p(n-1-j, x), for n >= 0, where E_x = x*d/dx (Euler operator). For the triangle a(n, m) this entails a recurrence for the sequence of column m, given in the formula section. - Wolfdieter Lang, Aug 11 2017
From Roger Ford, Jan 22 2018: (Start)
For row n, the nonzero values represent the odd components (loops) formed by n+1 nonintersecting arches above and below the x-axis with the following constraints:  The top has floor((n+3)/2) starting arches at position 1 and the next consecutive odd positions. All other starting top arches are in even positions. The bottom arches are a rainbow of arches.  If the component=1 then the arch configuration is a semimeander solution.
Examples: For row 3 {0, 2, 0, 1} there are 3 arch configurations: 2 arch configurations have a component=1; 1 has a component=3.  c=components, U=top arch starting in odd position, u=top arch starting in an even position, d=ending top arch:
.
  top UuUdUddd c=3   top UdUuUddd c=1     top UdUdUudd c=1
       /\                    /\
      //\\                  /  \
     //  \\                / /\ \                    /\
    //    \\              / /  \ \                  /  \
   ///\  /\\\        /\  / / /\ \ \        /\  /\  / /\ \
   \\\ \/ ///        \ \ \ \/ / / /        \ \ \ \/ / / /
    \\\  ///          \ \ \  / / /          \ \ \  / / /
     \\\///            \ \ \/ / /            \ \ \/ / /
      \\//              \ \  / /              \ \  / /
       \/                \ \/ /                \ \/ /
                          \  /                  \  /
                           \/                    \/
For row 4 {2, 0, 3, 0, 1} there are 6 arch configurations: 2 have a component=1; 3 have a component=3: 1 has a component=1. (End)

			Triangle a(n,m) begins:
  n\m  0   1   2   3   4   5   6  7  8  9 10 ...
  0:   1
  1:   0   1
  2:   1   0   1
  3:   0   2   0   1
  4:   2   0   3   0   1
  5:   0   5   0   4   0   1
  6:   5   0   9   0   5   0   1
  7:   0  14   0  14   0   6   0  1
  8:  14   0  28   0  20   0   7  0  1
  9:   0  42   0  48   0  27   0  8  0  1
  10: 42   0  90   0  75   0  35  0  9  0  1
  ... (Reformatted by _Wolfdieter Lang_, Sep 20 2013)
E.g., the fourth row corresponds to the polynomial p(3,x)= 2*x + x^3.
From _Paul Barry_, May 29 2009: (Start)
Production matrix is
  0, 1,
  1, 0, 1,
  0, 1, 0, 1,
  0, 0, 1, 0, 1,
  0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 0, 1 (End)
Boas-Buck recurrence for column k = 2, n = 6: a(6, 2) = (3/4)*(0 + 2*a(4 ,2) + 0 + 6*a(2, 2)) = (3/4)*(2*3 + 6) = 9. - _Wolfdieter Lang_, Aug 11 2017

J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, Ltd., Natick, MA, 2003. See p. 60. MR1957212 (2004a:17002)
A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
I. Bajunaid et al., Function series, Catalan numbers and random walks on trees, Amer. Math. Monthly 112 (2005), 765-785.
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 3.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Paul Barry and A. Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, example 3.
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.
J. Cigler, Some q-analogues of Fibonacci, Lucas and Chebyshev polynomials with nice moments, 2013.
J. Cigler, Some remarks about q-Chebyshev polynomials and q-Catalan numbers and related results, 2013.
J. Cigler, Some notes on q-Gould polynomials, 2013.
Emeric Deutsch, A. Robertson and D. Saracino, Refined restricted involutions, European Journal of Combinatorics 28 (2007), 481-498 (see pp. 486 and 498).
J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359, 2014
D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableau de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986 (see |F_{l,p}| on page 114). - _N. J. A. Sloane_, Jan 29 2011
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.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
Wolfdieter Lang, Chebyshev S-polynomials: ten applications.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Note 4, pp. 414-415.
MathOverflow, Catalan numbers as sums of squares of numbers in the rows of the Catalan triangle - is there a combinatorial explanation?
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Karim Ritter von Merkl, Computing colored Khovanov homology, arXiv:2505.03916 [math.QA], 2025. See p. 2.
Frank Ruskey and Mark Weston, Spherical Venn Diagrams with Involutory Isometries, Electronic Journal of Combinatorics, 18 (2011), #P191.
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Index entries for sequences related to Chebyshev polynomials.

Cf. A008315, A049310, A000108, A001405 (row sums), A145973, A153585, A108786, A037952. Another version: A008313. A039598 and A039599 without zeros, and odd and even numbered rows.

Variant without zero-diagonals: A033184 and with rows reversed: A009766.

a053121 n k = a053121_tabl !! n !! k
a053121_row n = a053121_tabl !! n
a053121_tabl = iterate
   (\row -> zipWith (+) ([0] ++ row) (tail row ++ [0,0])) [1]
-- Reinhard Zumkeller, Feb 24 2012

T:=proc(n,k): if n+k mod 2 = 0 then (k+1)*binomial(n+1,(n-k)/2)/(n+1) else 0 fi end: for n from 0 to 13 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Oct 12 2006
F:=proc(l,p) if ((l-p) mod 2) = 1 then 0 else (p+1)*l!/( ( (l-p)/2 )! * ( (l+p)/2 +1)! ); fi; end;
r:=n->[seq( F(n,p),p=0..n)]; [seq(r(n),n=0..15)]; # N. J. A. Sloane, Jan 29 2011
A053121 := proc(n,k) option remember; `if`(k>n or k<0,0,`if`(n=k,1,
procname(n-1,k-1)+procname(n-1,k+1))) end proc:
seq(print(seq(A053121(n,k), k=0..n)),n=0..12); # Peter Luschny, May 01 2011

a[n_, m_] /; n < m || OddQ[n-m] = 0; a[n_, m_] = (m+1) Binomial[n+1, (n-m)/2]/(n+1); Flatten[Table[a[n, m], {n, 0, 12}, {m, 0, n}]] [[1 ;; 90]] (* Jean-François Alcover, May 18 2011 *)
T[0, 0] := 1; T[n_, k_]/;0<=k<=n := T[n, k] = T[n-1, k-1]+T[n-1, k+1]; T[n_, k_] := 0; Flatten@Table[T[n, k], {n, 0, 12}, {k, 0, n}] (* Oliver Seipel, Dec 31 2024 *)

T(n, m)=if(nCharles R Greathouse IV, Mar 09 2016

def A053121_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1] + M[n-1,k+1]
    return M
A053121_triangle(13) # Peter Luschny, Sep 19 2012

a(n, m) := 0 if n

a(n, m) = (4*(n-1)*a(n-2, m) + 2*(m+1)*a(n-1, m-1))/(n+m+2), a(n, m)=0 if n

G.f. for m-th column: c(x^2)*(x*c(x^2))^m, where c(x) = g.f. for Catalan numbers A000108.

G.f.: G(t,z) = c(z^2)/(1 - t*z*c(z^2)), where c(z) = (1 - sqrt(1-4*z))/(2*z) is the g.f. for the Catalan numbers (A000108). - Emeric Deutsch, Jun 16 2011

a(n, m) = a(n-1, m-1) + a(n-1, m+1) if n > 0 and m >= 0, a(0, 0)=1, a(0, m)=0 if m > 0, a(n, m)=0 if m < 0. - Henry Bottomley, Jan 25 2001

Sum_{k>=0} T(m,k)^2 = A000108(m). - Paul D. Hanna, Apr 23 2005

Sum_{k>=0} T(m, k)*T(n, k) = 0 if m+n is odd; Sum_{k>=0} T(m, k)*T(n, k) = A000108((m+n)/2) if m+n is even. - Philippe Deléham, May 26 2005

T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i, C(i,j)*(C(i-j,j+k)-C(i-j,j+k+2))}}; Column k has e.g.f. BesselI(k,2x)-BesselI(k+2,2x). - Paul Barry, Feb 16 2006

Sum_{k=0..n} T(n,k)*(k+1) = 2^n. - Philippe Deléham, Mar 22 2007

Sum_{j>=0} T(n,j)*binomial(j,k) = A054336(n,k). - Philippe Deléham, Mar 30 2007

T(2*n+1,2*k+1) = A039598(n,k), T(2*n,2*k) = A039599(n,k). - Philippe Deléham, Apr 16 2007

Sum_{k=0..n} T(n,k)^x = A000027(n+1), A001405(n), A000108(n), A003161(n), A129123(n) for x = 0,1,2,3,4 respectively. - Philippe Deléham, Nov 22 2009

Sum_{k=0..n} T(n,k)*x^k = A126930(n), A126120(n), A001405(n), A054341(n), A126931(n) for x = -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Nov 28 2009

Sum_{k=0..n} T(n,k)*A000045(k+1) = A098615(n). - Philippe Deléham, Feb 03 2012

Recurrence for row polynomials C(n, x) := Sum_{m=0..n} a(n, m)*x^m = x*Sum_{k=0..n} Chat(k)*C(n-1-k, x), n >= 0, with C(-1, 1/x) = 1/x and Chat(k) = A000108(k/2) if n is even and 0 otherwise. From the o.g.f. of the row polynomials: G(z; x) := Sum_{n >= 0} C(n, x)*z^n = c(z^2)*(1 + x*z*G(z, x)), with the o.g.f. c of A000108. - Ahmet Zahid KÜÇÜK and Wolfdieter Lang, Aug 23 2015

The Boas-Buck recurrence (see a comment above) for the sequence of column m is: a(n, m) = ((m+1)/(n-m))*Sum_{j=0..n-1-m} (1/2)*(1 - (-1)^j)*binomial(j+1, (j+1)/2)* a(n-1-j, k), for n > m >= 0 and input a(m, m) = 1. - Wolfdieter Lang, Aug 11 2017

Sum_{m=1..n} a(n,m) = A037952(n). - R. J. Mathar, Sep 23 2021

Edited by N. J. A. Sloane, Jan 29 2011

A120730 Another version of Catalan triangle A009766.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 0, 5, 4, 1, 0, 0, 0, 5, 9, 5, 1, 0, 0, 0, 0, 14, 14, 6, 1, 0, 0, 0, 0, 14, 28, 20, 7, 1, 0, 0, 0, 0, 0, 42, 48, 27, 8, 1, 0, 0, 0, 0, 0, 42, 90, 75, 35, 9, 1, 0, 0, 0, 0, 0, 0, 132, 165, 110, 44, 10, 1

Offset: 0

Philippe Deléham, Aug 17 2006, corrected Sep 15 2006

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, ...] DELTA [1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, ...] where DELTA is the operator defined in A084938.
Aerated version gives A165408. - Philippe Deléham, Sep 22 2009
T(n,k) is the number of length n left factors of Dyck paths having k up steps. Example: T(5,4)=4 because we have UDUUU, UUDUU, UUUDU, and UUUUD, where U=(1,1) and D=(1,-1). - Emeric Deutsch, Jun 19 2011
With zeros omitted: 1,1,1,1,2,1,2,3,1,5,4,1,... = A008313. - Philippe Deléham, Nov 02 2011

			As a triangle, this begins:
  1;
  0,  1;
  0,  1,  1;
  0,  0,  2,  1;
  0,  0,  2,  3,  1;
  0,  0,  0,  5,  4,  1;
  0,  0,  0,  5,  9,  5,  1;
  0,  0,  0,  0, 14, 14,  6,  1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000007, A000096, A000108, A001405, A001477, A003161, A005586, A005587.
Cf. A008313, A009766, A039598, A039599, A084938, A099376, A105523, A121725.
Cf. A126087, A128386, A121724, A128387, A129123, A132373, A132374, A132375.
Cf. A132889, A151162, A151254, A151281, A156195, A156361, A156362, A156566.
Cf. A156577, A165408, A357654.

A120730:= func< n,k | n gt 2*k select 0 else Binomial(n, k)*(2*k-n+1)/(k+1) >;
[A120730(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Nov 07 2022

G := 4*z/((2*z-1+sqrt(1-4*z^2*t))*(1+sqrt(1-4*z^2*t))): Gser := simplify(series(G, z = 0, 13)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form  # Emeric Deutsch, Jun 19 2011
# second Maple program:
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
T:= (n, k)-> b(n, 2*k-n):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Oct 13 2022

b[x_, y_]:= b[x, y]= If[y<0 || y>x, 0, If[x==0, 1, Sum[b[x-1, y+j], {j, {-1, 1}}] ]];
T[n_, k_] := b[n, 2 k - n];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Oct 21 2022, after Alois P. Heinz *)
T[n_, k_]:= If[n>2*k, 0, Binomial[n, k]*(2*k-n+1)/(k+1)];
Table[T[n, k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 07 2022 *)

def A120730(n,k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
flatten([[A120730(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Nov 07 2022

G.f.: G(t,z) = 4*z/((2*z-1+sqrt(1-4*t*z^2))*(1+sqrt(1-4*t*z^2))). - Emeric Deutsch, Jun 19 2011
Sum_{k=0..n} x^k*T(n,n-k) = A001405(n), A126087(n), A128386(n), A121724(n), A128387(n), A132373(n), A132374(n), A132375(n), A121725(n) for x=1,2,3,4,5,6,7,8,9 respectively. [corrected by Philippe Deléham, Oct 16 2008]
T(2*n,n) = A000108(n); A000108: Catalan numbers.
From Philippe Deléham, Oct 18 2008: (Start)
Sum_{k=0..n} T(n,k)^2 = A000108(n) and Sum_{n>=k} T(n,k) = A000108(k+1).
Sum_{k=0..n} T(n,k)^3 = A003161(n).
Sum_{k=0..n} T(n,k)^4 = A129123(n). (End)
Sum_{k=0..n}, T(n,k)*x^k = A000007(n), A001405(n), A151281(n), A151162(n), A151254(n), A156195(n), A156361(n), A156362(n), A156566(n), A156577(n) for x=0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Feb 10 2009
From G. C. Greubel, Nov 07 2022: (Start)
T(n, k) = 0 if n > 2*k, otherwise binomial(n, k)*(2*k-n+1)/(k+1).
Sum_{k=0..n} (-1)^k*T(n,k) = A105523(n).
Sum_{k=0..n} (-1)^k*T(n,k)^2 = -A132889(n), n >= 1.
Sum_{k=0..floor(n/2)} T(n-k, k) = A357654(n).
T(n, n-1) = A001477(n).
T(n, n-2) = [n=2] + A000096(n-3), n >= 2.
T(n, n-3) = 2*[n<5] + A005586(n-5), n >= 3.
T(n, n-4) = 5*[n<7] - 2*[n=4] + A005587(n-7), n >= 4.
T(2*n+1, n+1) = A000108(n+1), n >= 0.
T(2*n-1, n+1) = A099376(n-1), n >= 1. (End)

A003161 A binomial coefficient sum.

Original entry on oeis.org

1, 1, 2, 9, 36, 190, 980, 5705, 33040, 204876, 1268568, 8209278, 53105976, 354331692, 2364239592, 16140234825, 110206067400, 765868074400, 5323547715200, 37525317999884, 264576141331216, 1886768082651816, 13458185494436592, 96906387191038334, 697931136204820336

Offset: 0

N. J. A. Sloane

The number of triples of standard tableaux of the same shape of height less than or equal to 2. - Mike Zabrocki, Mar 29 2007
From Peter Bala, Mar 20 2023: (Start)
For r a positive integer define S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r. The present sequence is {S(3,n)}. For other cases see A361887 ({S(5,n)}) and A361890 ({S(7,n)}).
Gould (1974) proposed the problem of showing that S(3,n) was always divisible by S(1,n). See A183069 for {S(3,n)/S(1,n)}. In fact, calculation suggests that if r is odd then S(r,n) is always divisible by S(1,n).
Conjecture: Let b(n) = a(2*n-1). Then the supercongruence b(n*p^k) == b(n*p^(k-1)) (mod p^(3*k)) holds for positive integers n and k and all primes p >= 5. (End)

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

Alois P. Heinz, Table of n, a(n) for n = 0..1116
F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.
H. W. Gould, Problem E2384, Amer. Math. Monthly, 81 (1974), 170-171.

Cf. A003162, A008315.
Cf. A001405, A000108, A129123, A183069.
Column k=3 of A357824.
Cf. A361887, A361890.

ogf := ((8*x-1)*(8*x+1)*hypergeom([1/4, 1/4],[1],64*x^2)^2/(x+1)-3*Int((16*x-5)*hypergeom([1/4, 1/4],[1],64*x^2)^2/(x+1)^2,x)+1)/(16*x);
series(ogf,x=0,30); # Mark van Hoeij, May 06 2013

Table[Sum[(Binomial[n, k]-Binomial[n, k-1])^3,{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Mar 06 2014 *)

a(n)=sum(k=0,n\2, (binomial(n,k)-binomial(n,k-1))^3) /* Michael Somos, Jun 02 2005 */

a(n) = Sum_{k=0..n} A120730(n,k)^3. - Philippe Deléham, Oct 18 2008
G.f.: hypergeometric expression with an anti-derivative, see Maple program. - Mark van Hoeij, May 06 2013
Recurrence: n*(n+1)^3*(7*n^2 - 14*n + 3)*a(n) = - n*(7*n^5 - 112*n^4 + 206*n^3 + 8*n^2 - 125*n + 48)*a(n-1) + 16*(n-1)*(28*n^5 - 133*n^4 + 194*n^3 - 33*n^2 - 120*n + 61)*a(n-2) + 64*(n-2)^3*(n-1)*(7*n^2 - 4)*a(n-3). - Vaclav Kotesovec, Mar 06 2014
a(n) ~ 2^(3*n+9/2) / (9 * Pi^(3/2) * n^(5/2)). - Vaclav Kotesovec, Mar 06 2014
a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^3. - Alois P. Heinz, Oct 17 2022

A357824 Total number A(n,k) of k-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 5, 6, 3, 1, 1, 2, 9, 14, 10, 4, 1, 1, 2, 17, 36, 42, 20, 4, 1, 1, 2, 33, 98, 190, 132, 35, 5, 1, 1, 2, 65, 276, 882, 980, 429, 70, 5, 1, 1, 2, 129, 794, 4150, 7812, 5705, 1430, 126, 6, 1, 1, 2, 257, 2316, 19722, 65300, 78129, 33040, 4862, 252, 6

Offset: 0

Alois P. Heinz, Oct 14 2022

			Square array A(n,k) begins:
  1,  1,   1,    1,     1,       1,        1,         1, ...
  1,  1,   1,    1,     1,       1,        1,         1, ...
  2,  2,   2,    2,     2,       2,        2,         2, ...
  2,  3,   5,    9,    17,      33,       65,       129, ...
  3,  6,  14,   36,    98,     276,      794,      2316, ...
  3, 10,  42,  190,   882,    4150,    19722,     94510, ...
  4, 20, 132,  980,  7812,   65300,   562692,   4939220, ...
  4, 35, 429, 5705, 78129, 1083425, 15105729, 211106945, ...

Alois P. Heinz, Antidiagonals n = 0..121, flattened
Wikipedia, Counting lattice paths

Columns k=0-7 give A008619, A001405, A000108, A003161, A129123, A361887, A382433, A361890.
Rows n=1-5 give: A000012, A007395, A000051, A001550, A074511.
Main diagonal gives A357825.
Cf. A008315, A120730.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
A:= (n, k)-> add(b(n, n-2*j)^k, j=0..n/2):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - 1, y + j], {j, {-1, 1}}]]];
A[n_, k_] := Sum[b[n, n - 2*j]^k, { j, 0, n/2}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Oct 18 2022, after Alois P. Heinz *)

A(n,k) = Sum_{j=0..floor(n/2)} A008315(n,j)^k.

A(n,k) = Sum_{j=0..n} A120730(n,j)^k for k>=1, A(n,0) = A008619(n).

A382433 a(n) = S(6,n), where S(r,n) = Sum_{k=0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

Original entry on oeis.org

1, 1, 2, 65, 794, 19722, 562692, 15105729, 553537490, 18107304842, 716747344436, 27247858130506, 1137502720488532, 47573235297987700, 2085487143991309320, 92820152112054862785, 4246321874111740074210, 197525644801830489637170, 9363425291004877645851300

Offset: 0

Seiichi Manyama, Mar 25 2025

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..562

Column k=6 of A357824.
Cf. A000108, A129123.
Cf. A008315, A120730, A382435.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> add(b(n, n-2*j)^6, j=0..n/2):
seq(a(n), n=0..18);  # Alois P. Heinz, Mar 25 2025

Table[Sum[Binomial[n,k] * (Binomial[n,k] - Binomial[n,k-1])^5, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 25 2025 *)

a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^5);

from math import comb
def A382433(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**6 for j in range((n>>1)+1)) # Chai Wah Wu, Mar 25 2025

a(n) = Sum_{k=0..floor(n/2)} A008315(n,k)^6.
a(n) = Sum_{k=0..n} A120730(n,k)^6.
a(n) = A357824(n,6).
a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^5.
a(n) ~ 5 * 2^(6*n+4) / (3^(5/2) * Pi^(5/2) * n^(11/2)). - Vaclav Kotesovec, Mar 25 2025

A355481 Number of pairs of Dyck paths of semilength n such that the midpoint of the first is above the midpoint of the second.

Original entry on oeis.org

0, 0, 1, 4, 49, 441, 4806, 52956, 614713, 7341697, 90118054, 1130414649, 14447230854, 187609607862, 2470253990556, 32922380442828, 443493622670313, 6031353319151961, 82725531355436886, 1143385727109903585, 15913217995801644870, 222875331740976566070

Offset: 0

Alois P. Heinz, Oct 07 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..839
Wikipedia, Counting lattice paths

Cf. A000108, A001246, A129123, A357652.

b:= proc(n) option remember; `if`(n<2, 1, (2*n*(90*n^5-309*n^4+147*n^3+
      124*n^2-135*n+35)*b(n-1)+4*(n-1)^2*(4*n-5)*(4*n-3)*(15*n^2-4*n-12)*
       b(n-2))/(n*(n+1)^3*(15*n^2-34*n+7)))
    end:
a:= n-> ((binomial(n+n, n)/(n+1))^2-b(n))/2:
seq(a(n), n=0..21);

A129123[n_] := Sum[(Binomial[n, k]-Binomial[n, k-1])^4, {k, 0, Floor[n/2]}];
a[n_] := (CatalanNumber[n]^2 - A129123[n])/2;
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 16 2022 *)

a(n) = (A001246(n) - A129123(n))/2 = (A000108(n)^2 - A129123(n))/2.

A357652 Number of pairs of Dyck paths of semilength n such that the midpoint of the first is not below the midpoint of the second.

Original entry on oeis.org

1, 1, 3, 21, 147, 1323, 12618, 131085, 1430187, 16297347, 191987562, 2325379147, 28821761290, 364290802138, 4682375323044, 61067639131197, 806671205158587, 10776418254992139, 145413196382253114, 1979833455619072515, 27174458892459331530, 375722890152963114330

Offset: 0

Alois P. Heinz, Oct 07 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..839
Wikipedia, Counting lattice paths

Cf. A000108, A001246, A129123, A355481.

b:= proc(n) option remember; `if`(n<2, 1, (2*n*(90*n^5-309*n^4+147*n^3+
      124*n^2-135*n+35)*b(n-1)+4*(n-1)^2*(4*n-5)*(4*n-3)*(15*n^2-4*n-12)*
       b(n-2))/(n*(n+1)^3*(15*n^2-34*n+7)))
    end:
a:= n-> ((binomial(n+n, n)/(n+1))^2+b(n))/2:
seq(a(n), n=0..21);

a(n) = (A001246(n) + A129123(n))/2 = (A000108(n)^2 + A129123(n))/2.

A381676 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^2.

Original entry on oeis.org

1, 1, 4, 17, 86, 472, 2752, 16753, 105394, 680366, 4484360, 30067160, 204508240, 1408057120, 9796738304, 68786005361, 486845236106, 3470187822754, 24891491746792, 179556655434382, 1301857088258836, 9482632068303296, 69361538748381824, 509303099950899352

Offset: 0

Seiichi Manyama, Mar 25 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000108, A129123, A382433, A382443, A382446.
Cf. A131428, A178824.
Cf. A000172, A086618, A238115.

[ &+[Binomial(n, k)^2 * (Binomial(n, k) - (k gt 0 select Binomial(n, k-1) else 0)) : k in [0..n]] : n in [0..20] ]; // Vincenzo Librandi, Mar 27 2025

Table[Sum[Binomial[n,k]^2*(Binomial[n,k]-Binomial[n,k-1]),{k,0,n}],{n,0,20}] (* Vincenzo Librandi, Mar 27 2025 *)

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

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

a(n) ~ 2^(3*n+3) / (Pi * 3^(3/2) * n^2). - Vaclav Kotesovec, Mar 26 2025

A382434 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^4.

Original entry on oeis.org

1, 1, 3, 33, 195, 1763, 15623, 156257, 1630947, 17911299, 203739015, 2389928995, 28749060871, 353362388551, 4424242664975, 56290517376737, 726355164976547, 9490129871680355, 125375330053632455, 1672895457018337859, 22522481793315373319, 305695116823973096519

Offset: 0

Seiichi Manyama, Mar 25 2025

nonn

Cf. A131428, A382435.

Cf. A080233, A129123, A156644.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> 2*add(b(n, n-2*j)^4, j=0..n/2)-1:
seq(a(n), n=0..21);  # Alois P. Heinz, Mar 25 2025

a(n) = sum(k=0, n, (binomial(n, k)-binomial(n, k-1))^4);

from math import comb
def A382434(n): return (sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**4 for j in range((n>>1)+1))<<1)-1 # Chai Wah Wu, Mar 25 2025

a(n) = Sum_{k=0..n} A080233(n,k)^4 = Sum_{k=0..n} A156644(n,k)^4.
a(n) = 2 * A129123(n) - 1.
D-finite with recurrence n*(n+1)^3*a(n) -2*n*(11*n^3-17*n^2+5*n+5)*a(n-1) -4*(n-1)*(70*n^3-365*n^2+527*n-162)*a(n-2) +8*(n-2)*(584*n^3-5020*n^2+14111*n-13059)*a(n-3) +1344*(4*n-11)*(4*n-13)*(-3+n)^2*a(n-4) +9*(2875*n^4-33975*n^3+149945*n^2-293541*n+215336)=0. - R. J. Mathar, Mar 31 2025

A382443 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^4.

Original entry on oeis.org

1, 1, 4, 65, 566, 10912, 164032, 3237313, 62253130, 1314421886, 28392213224, 639799858304, 14785604868256, 350615631856960, 8485316740880384, 209179475361783233, 5239271305444731698, 133100429387161703962, 3424142506153260211720, 89090362800169426107070

Offset: 0

Seiichi Manyama, Mar 26 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A000108, A129123, A381676, A382433, A382446.

Cf. A382434.

[&+[Binomial(n, k)* (Binomial(n, k) - Binomial (n, k-1))^4: k in [0..n]]: n in [0..21]]; // Vincenzo Librandi, Mar 29 2025

Table[Sum[Binomial[n,k]*(Binomial[n,k]-Binomial[n,k-1])^4,{k,0,n}],{n,0,20}] (* Vincenzo Librandi, Mar 29 2025 *)

a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^4);

a(n) = Sum_{k=0..n} binomial(n,k)^2 * ( binomial(n,k) - binomial(n,k-1) )^3.

a(n) ~ 3 * 2^(5*n+6) / (Pi^2 * 5^(5/2) * n^4). - Vaclav Kotesovec, Mar 26 2025

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cp's OEIS Frontend

A053121 Catalan triangle (with 0's) read by rows.

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A120730 Another version of Catalan triangle A009766.

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A003161 A binomial coefficient sum.

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A357824 Total number A(n,k) of k-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A382433 a(n) = S(6,n), where S(r,n) = Sum_{k=0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

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A355481 Number of pairs of Dyck paths of semilength n such that the midpoint of the first is above the midpoint of the second.

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A357652 Number of pairs of Dyck paths of semilength n such that the midpoint of the first is not below the midpoint of the second.

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