A054341 -id:A054341 - 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?
A. Nkwanta and A. Tefera, Curious Relations and Identities Involving the Catalan Generating Function and Numbers, Journal of Integer Sequences, 16 (2013), #13.9.5.
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.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
Yidong Sun and Luping Ma, Minors of a class of Riordan arrays related to weighted partial Motzkin paths. Eur. J. Comb. 39, 157-169 (2014).
Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes.
W.-J. Woan, Area of Catalan Paths, Discrete Math., 226 (2001), 439-444.
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

A126075 Triangle T(n,k), 0 <= k <= n, read by rows, defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 2*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k+1) for k >= 1.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 12, 6, 2, 1, 30, 14, 7, 2, 1, 74, 37, 16, 8, 2, 1, 185, 90, 45, 18, 9, 2, 1, 460, 230, 108, 54, 20, 10, 2, 1, 1150, 568, 284, 128, 64, 22, 11, 2, 1, 2868, 1434, 696, 348, 150, 75, 24, 12, 2, 1

Offset: 0

Philippe Deléham, Mar 02 2007

Riordan array (c(x^2)/(1-2xc(x^2)),xc(x^2)) where c(x)=g.f. of Catalan numbers A000108. - Philippe Deléham, Mar 18 2007

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

			Triangle begins:
     1;
     2,    1;
     5,    2,   1;
    12,    6,   2,   1;
    30,   14,   7,   2,   1;
    74,   37,  16,   8,   2,  1;
   185,   90,  45,  18,   9,  2,  1;
   460,  230, 108,  54,  20, 10,  2,  1;
  1150,  568, 284, 128,  64, 22, 11,  2, 1;
  2868, 1434, 696, 348, 150, 75, 24, 12, 2, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
P. Bala, A 4-parameter family of embedded Riordan arrays

Cf. A000108, A054341, A127358.

A126075 := proc (n, k)
add( 2^(n-k-2*j)*binomial(n, j), j = 0..floor((n-k)/2) ) - add( 2^(n-k-2-2*j)*binomial(n, j), j = 0..floor((n-k-2)/2) )
end proc:
# display sequence in triangular form
for n from 0 to 10 do seq(A126075(n, k), k = 0..n) end do;
# Peter Bala, Feb 20 2018

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]]; Table[T[n, k, 2, 0], {n, 0, 49}, {k, 0, n}] // Flatten  (* G. C. Greubel, Apr 21 2017 *)

Sum_{k=0..n} T(n,k) = A127358(n). T(n,0)=A054341(n).
Sum_{k=0..n} T(n,k)*(-k+1) = 2^n. - Philippe Deléham, Mar 25 2007
From Peter Bala, Feb 20 2018: (Start)
T(n,k) = Sum_{j = 0..floor((n-k)/2)} 2^(n-k-2*j)*binomial(n, j) - Sum_{j = 0..floor((n-k-2)/2)} 2^(n-k-2-2*j)*binomial(n, j), 0 <= k <= n. - Peter Bala, Feb 20 2018
The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 - x^2)/(1 - 2*x) * (1 + x^2)^n about 0. For example, for n = 4, (1 - x^2)/(1 - 2*x) * (1 + x^2)^4 = (30*x^4 + 14*x*3 + 7*x^2 + 2*x + 1) + O(x^5). (End)

A054336 A convolution triangle of numbers based on A001405 (central binomial coefficients).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 6, 10, 9, 4, 1, 10, 22, 22, 14, 5, 1, 20, 44, 54, 40, 20, 6, 1, 35, 93, 123, 109, 65, 27, 7, 1, 70, 186, 281, 276, 195, 98, 35, 8, 1, 126, 386, 618, 682, 541, 321, 140, 44, 9, 1, 252, 772, 1362, 1624, 1440, 966, 497, 192, 54, 10, 1

Offset: 0

Wolfdieter Lang, Mar 13 2000

T(n,k) is the number of 2-Motzkin paths (i.e., Motzkin paths with blue and red level steps) with no level steps at positive height and having k blue level steps. Example: T(4,2)=9 because, denoting U=(1,1), D=(1,-1), B=blue (1,0), R=red (1,0), we have BBRR, BRBR, BRRB, RBBR, RBRB, RRBB, BBUD, BUDB, and UDBB. - Emeric Deutsch, Jun 07 2011
In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The g.f. for the row polynomials p(n,x) (increasing powers of x) is 1/(1-(1+x)*z-z^2*c(z^2)), with c(x) the g.f. for Catalan numbers A000108.
Column sequences: A001405, A045621.
Riordan array (f(x), x*f(x)), f(x) the g.f. of A001405. - Philippe Deléham, Dec 08 2009
From Paul Barry, Oct 21 2010: (Start)
Riordan array ((sqrt(1+2x) - sqrt(1-2x))/(2x*sqrt(1-2x)), (sqrt(1+2x)-sqrt(1-2x))/(2*sqrt(1-2x))),
inverse of Riordan array ((1+x)/(1+2x+2x^2), x(1+x)/(1+2x+2x^2)) (A181472). (End)

			Fourth row polynomial (n=3): p(3,x)= 3 + 5*x + 3*x^2 + x^3.
From _Paul Barry_, Oct 21 2010: (Start)
Triangle begins
   1;
   1,  1;
   2,  2,   1;
   3,  5,   3,   1;
   6, 10,   9,   4,  1;
  10, 22,  22,  14,  5,  1;
  20, 44,  54,  40, 20,  6, 1;
  35, 93, 123, 109, 65, 27, 7, 1;
Production matrix is
   1,  1;
   1,  1,  1;
  -1,  1,  1,  1;
   1, -1,  1,  1,  1;
  -1,  1, -1,  1,  1,  1;
   1, -1,  1, -1,  1,  1,  1;
  -1,  1, -1,  1, -1,  1,  1, 1;
   1, -1,  1, -1,  1, -1,  1, 1, 1;
  -1,  1, -1,  1, -1,  1, -1, 1, 1, 1; (End)

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A001405, A035324, A054335.

Row sums: A054341.

A053121:= function(n,k)
    if ((n-k+1) mod 2)=0 then return 0;
    else return (k+1)*Binomial(n+1, Int((n-k)/2))/(n+1);
    fi;
  end;
T:= function(n,k)
    return Sum([k..n], j-> Binomial(j,k)*A053121(n,j));
  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jul 21 2019

A053121:= func< n,k | ((n-k+1) mod 2) eq 0 select 0 else (k+1)*Binomial(n+1, Floor((n-k)/2))/(n+1) >;
T:= func< n,k | (&+[Binomial(j,k)*A053121(n,j): j in [k..n]]) >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 21 2019

c[n_, j_] /; n < j || OddQ[n - j] = 0; c[n_, j_] = (j + 1) Binomial[n + 1, (n - j)/2]/(n + 1); t[n_, k_] := Sum[c[n, j]*Binomial[j, k], {j, 0, n}]; Flatten[Table[t[n, k], {n, 0, 10}, {k, 0, n}]][[;; 66]] (* Jean-François Alcover, Jul 13 2011, after Philippe Deléham *)

A053121(n,k) = if((n-k+1)%2==0, 0, (k+1)*binomial(n+1, (n-k)\2)/(n+1) );
T(n,k) = sum(j=k,n, A053121(n,j)*binomial(j,k));
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 21 2019

def A053121(n, k):
    if (n-k+1) % 2==0: return 0
    else: return (k+1)*binomial(n+1, ((n-k)//2))/(n+1)
def T(n,k): return sum(binomial(j,k)*A053121(n,j) for j in (k..n))
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 21 2019

G.f. for column m: cbi(x)*(x*cbi(x))^m, with cbi(x) := (1+x*c(x^2))/sqrt(1-4*x^2) = 1/(1-x-x^2*c(x^2)), where c(x) is the g.f. for Catalan numbers A000108.
T(n,k) = Sum_{j>=0} A053121(n,j)*binomial(j,k). - Philippe Deléham, Mar 30 2007
T(n,k) = T(n-1,k-1) + T(n-1,l) + Sum_{j>=0} T(n-1,k+1+j)*(-1)^j. - Philippe Deléham, Feb 23 2012

A127358 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*2^(n-k).

Original entry on oeis.org

1, 3, 8, 21, 54, 138, 350, 885, 2230, 5610, 14088, 35346, 88596, 221952, 555738, 1391061, 3480870, 8708610, 21783680, 54483510, 136254964, 340729788, 852000828, 2130354786, 5326563004, 13317759588, 33296999120, 83247698100, 208129274400, 520343244300

Offset: 0

Paul Barry, Jan 11 2007

Hankel transform is (-1)^n. In general, given r >= 0, the sequence given by Sum_{k=0..n} binomial(n, floor(k/2))*r^(n-k) has Hankel transform (1-r)^n. The sequence is the image of the sequence with g.f. (1+x)/(1-2*x) under the Chebyshev mapping g(x) -> (1/sqrt(1-4*x^2))*g(x*c(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.

			a(3) = 21 = (12 + 6 + 2 + 1), where the top row of M^3 = (12, 6, 2, 1).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Cf. A107430. - Philippe Deléham, Sep 16 2009

Table[Sum[Binomial[n,Floor[k/2]]2^(n-k),{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jun 03 2012 *)
CoefficientList[Series[(1 + 2*x - Sqrt[1 - 4*x^2])/(2*Sqrt[1 - 4*x^2]*(x - 1 + Sqrt[1 - 4*x^2])), {x, 0, 50}], x] (* G. C. Greubel, May 22 2017 *)

my(x='x+O('x^50)); Vec((1 + 2*x - sqrt(1 - 4*x^2))/(2*sqrt(1 - 4*x^2)*(x - 1 + sqrt(1 - 4*x^2)))) \\ G. C. Greubel, May 22 2017

G.f.: (1/sqrt(1 - 4x^2))(1 + x*c(x^2))/(1 - 2*x*c(x^2)).
a(n) = 2*a(n-1) + A054341(n-1). a(n) = Sum_{k=0..n} A126075(n,k). - Philippe Deléham, Mar 03 2007
a(n) = Sum_{k=0..n} A061554(n,k)*2^k. - Philippe Deléham, Dec 04 2009
From Gary W. Adamson, Sep 07 2011: (Start)
a(n) is the sum of top row terms of M^n, M is an infinite square production matrix as follows:
  2, 1, 0, 0, 0, ...
  1, 0, 1, 0, 0, ...
  0, 1, 0, 1, 0, ...
  0, 0, 1, 0, 1, ...
  0, 0, 0, 1, 0, ...
  ... (End)
D-finite with recurrence 2*n*a(n) + (-5*n-4)*a(n-1) + 2*(-4*n+13)*a(n-2) + 20*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 30 2012
a(n) ~ 3 * 5^n / 2^(n+1). - Vaclav Kotesovec, Feb 13 2014

A305561 Expansion of 2x(1 - 2x)/(1 + 2x - 8x^2 - sqrt(1 - 4x^2)).

Original entry on oeis.org

1, 1, 3, 8, 23, 64, 182, 512, 1451, 4096, 11594, 32768, 92710, 262144, 741548, 2097152, 5931955, 16777216, 47454210, 134217728, 379628818, 1073741824, 3037013748, 8589934592, 24296051198, 68719476736, 194368201572, 549755813888, 1554944869676, 4398046511104

Offset: 0

Ilya Gutkovskiy, Jun 21 2018

nonn

Invert transform of A001405.

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Central Binomial Coefficient

Cf. A001405, A026671, A054341, A075436, A293732, A293741.

m:=35; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 2*x*(1 - 2*x)/(1 + 2*x - 8*x^2 - Sqrt(1 - 4*x^2)))); // Vincenzo Librandi, Jan 27 2020

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-i)*binomial(i, floor(i/2)), i=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 21 2018

nmax = 29; CoefficientList[Series[2 x (1 - 2 x)/(1 + 2 x - 8 x^2 - Sqrt[1 - 4 x^2]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[Binomial[k, Floor[k/2]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[k, Floor[k/2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]

G.f.: 1/(1 - Sum_{k>=1} binomial(k,floor(k/2))*x^k).
D-finite with recurrence: n*(n+1)*a(n) +(n-1)*(n-5)*a(n-1) -12*(n-1)*(n+1)*a(n-2) -12*(n-2)*(n-5)*a(n-3) +32*(n+1)*(n-3)*a(n-4) +32*(n-4)*(n-5)*a(n-5)=0. - R. J. Mathar, Jan 25 2020
a(n) ~ 2^(3*(n-1)/2). - Vaclav Kotesovec, Jan 29 2020

A369432 Number of Dyck excursions with catastrophes from (0,0) to (n,0).

Original entry on oeis.org

1, 1, 3, 6, 16, 37, 95, 230, 582, 1434, 3606, 8952, 22446, 55917, 140007, 349374, 874150, 2183230, 5460506, 13643972, 34118328, 85270626, 213205958, 532926716, 1332420796, 3330739972, 8327221380, 20816939100, 52043684970, 130105200765, 325267849335, 813155081070

Offset: 0

Florian Schager, Jan 23 2024

A Dyck excursion is a lattice path with steps U = (1,1) and D = (1,-1) that does not go below the x-axis and ends at the x-axis.

A catastrophe is a step C = (1,-k) from altitude k to altitude 0 for k >= 0.

			For n = 3 the a(3) = 6 solutions are UUC, UDC, UCC, CUD, CUC, CCC.
For n = 4 the a(4) = 16 solutions are UUUC, UUDD, UUDC, UUCC, UDUD, UDUC, UDCC, UCUD, UCUC, UCCC, CUUC, CUDC, CUCC, CCUD, CCUC, CCCC.

Alois P. Heinz, Table of n, a(n) for n = 0..2514
Cyril Banderier and Michael Wallner, Lattice paths with catastrophes, arXiv:1707.01931 [math.CO], 2017, p.7.

Cf. A054341 (Dyck meanders with catastrophes).

Cf. A224747 (different model of catastrophes).

u1 := solve(1 - z*(1/u + u), u)[2];
M := (1 - u1)/(1 - 2*z);
E := u1/z;
F := E/(-M*z + 1);
series(F, z, 33);
# second Maple program:
b:= proc(x, y) option remember; `if`(x=0, `if`(y=0, 1, 0),
      b(x-1, 0)+`if`(y>0, b(x-1, y-1), 0)+b(x-1, y+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..31);  # Alois P. Heinz, Jan 23 2024

b[x_, y_] := b[x, y] = If[x == 0, If[y == 0, 1, 0], b[x-1, 0] + If[y > 0, b[x-1, y-1], 0] + b[x-1, y+1]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jul 24 2025, after Alois P. Heinz *)

my(N=44,z='z+O('z^N)); Vec((1 - sqrt(1 -4*z^2))*(2*z - 1)/(z^2*(6*z - 3 + sqrt(1 - 4*z^2))))

G.f.: (1 - sqrt(1 - 4*z^2))*(2*z - 1)/(z^2*(6*z - 3 + sqrt(1 - 4*z^2))).

a(n) ~ 3/8*(5/2)^n.

A171388 Expansion of the first column of triangle T_(2,x), T_(x,y) defined in A039599; T_(2,0)= A126075, T_(2,1)= A038622, T_(2,2)= A039598, T_(2,3)= A124733, T_(2,4)= A124575.

Original entry on oeis.org

1, 2, 0, 5, 0, 0, 12, 1, 0, 0, 30, 4, 1, 0, 0, 74, 17, 4, 1, 0, 0, 185, 56, 21, 4, 1, 0, 0, 460, 185, 74, 26, 4, 1, 0, 0

Offset: 0

Philippe Deléham, Dec 07 2009

			Triangle begins:
   1;
   2,  0;
   5,  0, 0;
  12,  1, 0, 0;
  30,  4, 1, 0, 0;
  74, 17, 4, 1, 0, 0;
  ...

Cf. A000108, A171368, A171380.

Sum_{k=0..n} T(n,k)*x^k = A054341(n), A005773(n+1), A000108(n+1), A007317(n), A033543(n) for x = 0, 1, 2, 3, 4 respectively.

A171509 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A126931.

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 33, 29, 9, 1, 110, 126, 57, 12, 1, 366, 518, 306, 94, 15, 1, 1220, 2052, 1494, 600, 140, 18, 1, 4065, 7925, 6849, 3389, 1035, 195, 21, 1, 13550, 30030, 30025, 17628, 6635, 1638, 259, 24, 1

Offset: 0

Philippe Deléham, Dec 10 2009

Equal to A053121*B^3, B = A007318.

			Triangle begins:
  1 ;
  3,1 ;
  10,6,1 ;
  33,29,9,1 ;
  110,126,57,12,1 ; ...

Cf. A053121, A054336, A096164, A171505.

Sum_{k=0..n} T(n,k)*x^k = A126930(n), A126120(n), A001405(n), A054341(n), A126931(n) for x = -4, -3, -2, -1, 0 respectively.

T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + Sum_{i>=0} T(n-1,k+1+i)*(-3)^i. - Philippe Deléham, Feb 23 2012

A171616 Triangle T : T(n,k)= binomial(n,k)*A000957(n+1-k).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 6, 8, 6, 0, 1, 18, 30, 20, 10, 0, 1, 57, 108, 90, 40, 15, 0, 1, 186, 399, 378, 210, 70, 21, 0, 1, 622, 1488, 1596, 1008, 420, 112, 28, 0, 1, 2120, 5598, 6696, 4788, 2268, 756, 168, 36, 0, 1, 7338, 21200, 27990, 22320, 11970, 4536, 1260, 240, 45

Offset: 0

Philippe Deléham, Dec 13 2009

			Triangle begins : 1 ; 0,1 ; 1,0,1 ; 2,3,0,1 ; 6,8,6,0,1 ; 18,30,20,10,0,1 ; ...

Cf. A007318

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000957(n+1), A033321(n), A033543(n) for x = 0,1,2 respectively. Sum_{k, 0<=k<=n} T(n,k)*(-1)^(n-k)*x^k = A054341(n), A059738(n), A049027(n+1) for x = 2,3,4 respectively.

Showing 1-10 of 12 results. Next

cp's OEIS Frontend

A059738 Binomial transform of A054341 and inverse binomial transform of A049027.

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

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A126075 Triangle T(n,k), 0 <= k <= n, read by rows, defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 2*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k+1) for k >= 1.

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A054336 A convolution triangle of numbers based on A001405 (central binomial coefficients).

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A127358 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*2^(n-k).

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A305561 Expansion of 2*x*(1 - 2*x)/(1 + 2*x - 8*x^2 - sqrt(1 - 4*x^2)).

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A305561 Expansion of 2x(1 - 2x)/(1 + 2x - 8x^2 - sqrt(1 - 4x^2)).