A128899 -id:A128899 - cp's OEIS frontend

A382394 a(n) = Sum_{k=0..n} A128899(n,k)^3.

1, 1, 9, 190, 5705, 204876, 8209278, 354331692, 16140234825, 765868074400, 37525317999884, 1886768082651816, 96906387191038334, 5066711735118128200, 268954195756648761900, 14464077426547576156440, 786729115199980286001225, 43219452658242723841261800

Offset: 0

Seiichi Manyama, Mar 24 2025

nonn

Let b_k(n) = Sum_{j=0..n} A128899(n,j)^k. b_1(n) = binomial(2*n-1,n) = A088218(n) and b_2(n) = A024492(n-1) for n > 0.

Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, Sums of powers of Catalan triangle numbers, arXiv:1602.04347 [math.NT], 2016.

Cf. A024492, A088218.

Cf. A001700, A003161, A128899, A183069.

a128899(n, k) = binomial(2*n-2, n-k)-binomial(2*n-2, n-k-2);
a(n) = sum(k=0, n, a128899(n, k)^3);

a(n) = binomial(2*n,n)/2 * A183069(n) for n > 0.

a(n) = A003161(2*n-1) for n > 0.

A359108 a(n) = A128899(2n, n) = 2binomial(4n - 1, 3n) for n >= 1 and a(0) = 1.

Original entry on oeis.org

1, 2, 14, 110, 910, 7752, 67298, 592020, 5259150, 47071640, 423830264, 3834669566, 34834267234, 317506779800, 2902365981900, 26597044596360, 244263468539790, 2247575790712824, 20716044882791720, 191230475831922200, 1767658071106087160, 16359617358545329440

Offset: 0

Peter Luschny, Dec 27 2022

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A005810, A128899, A224274.

a := n -> ifelse(n = 0, 1, 2*binomial(4*n - 1, 3*n)):
# Alternative:
a := proc(n) option remember; ifelse(n < 2, n + 1, (8*(2*n - 1) * (4*n - 3) * (4*n - 1) * a(n - 1)) / (3 * n * (3*n - 2) * (3*n - 1))) end:
seq(a(n), n = 0..19);

A359108[n_] := If[n == 0, 1, 2*Binomial[4*n - 1, 3*n]];
Array[A359108, 25, 0] (* Paolo Xausa, Sep 18 2024 *)

a(n) = (8*(2*n - 1) * (4*n - 3) * (4*n - 1) * a(n - 1)) / (3*n * (3*n - 2) * (3*n - 1)) for n >= 2.
a(n) = (1/2)*A005810(n) = 2*A224274(n) for n >= 1. - Peter Bala, Feb 08 2023
a(n) = [x^n] C(x)^(2*n), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Apr 27 2023

A039599 Triangle formed from even-numbered columns of triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 9, 5, 1, 14, 28, 20, 7, 1, 42, 90, 75, 35, 9, 1, 132, 297, 275, 154, 54, 11, 1, 429, 1001, 1001, 637, 273, 77, 13, 1, 1430, 3432, 3640, 2548, 1260, 440, 104, 15, 1, 4862, 11934, 13260, 9996, 5508, 2244, 663, 135, 17, 1

Offset: 0

N. J. A. Sloane

T(n,k) is the number of lattice paths from (0,0) to (n,n) with steps E = (1,0) and N = (0,1) which touch but do not cross the line x - y = k and only situated above this line; example: T(3,2) = 5 because we have EENNNE, EENNEN, EENENN, ENEENN, NEEENN. - Philippe Deléham, May 23 2005
The matrix inverse of this triangle is the triangular matrix T(n,k) = (-1)^(n+k)* A085478(n,k). - Philippe Deléham, May 26 2005
Essentially the same as A050155 except with a leading diagonal A000108 (Catalan numbers) 1, 1, 2, 5, 14, 42, 132, 429, .... - Philippe Deléham, May 31 2005
Number of Grand Dyck paths of semilength n and having k downward returns to the x-axis. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)). Example: T(3,2)=5 because we have u(d)uud(d),uud(d)u(d),u(d)u(d)du,u(d)duu(d) and duu(d)u(d) (the downward returns to the x-axis are shown between parentheses). - Emeric Deutsch, May 06 2006
Riordan array (c(x),x*c(x)^2) where c(x) is the g.f. of A000108; inverse array is (1/(1+x),x/(1+x)^2). - Philippe Deléham, Feb 12 2007
The triangle may also be generated from M^n*[1,0,0,0,0,0,0,0,...], where M is the infinite tridiagonal matrix with all 1's in the super and subdiagonals and [1,2,2,2,2,2,2,...] in the main diagonal. - Philippe Deléham, Feb 26 2007
Inverse binomial matrix applied to A124733. Binomial matrix applied to A089942. - Philippe Deléham, Feb 26 2007
Number of standard tableaux of shape (n+k,n-k). - Philippe Deléham, Mar 22 2007
From Philippe Deléham, Mar 30 2007: (Start)
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. (End)
The table U(n,k) = Sum_{j=0..n} T(n,j)*k^j is given in A098474. - Philippe Deléham, Mar 29 2007
Sequence read mod 2 gives A127872. - Philippe Deléham, Apr 12 2007
Number of 2n step walks from (0,0) to (2n,2k) and consisting of step u=(1,1) and d=(1,-1) and the path stays in the nonnegative quadrant. Example: T(3,0)=5 because we have uuuddd, uududd, ududud, uduudd, uuddud; T(3,1)=9 because we have uuuudd, uuuddu, uuudud, ududuu, uuduud, uduudu, uudduu, uduuud, uududu; T(3,2)=5 because we have uuuuud, uuuudu, uuuduu, uuduuu, uduuuu; T(3,3)=1 because we have uuuuuu. - Philippe Deléham, Apr 16 2007, Apr 17 2007, Apr 18 2007
Triangular matrix, read by rows, equal to the matrix inverse of triangle A129818. - Philippe Deléham, Jun 19 2007
Let Sum_{n>=0} a(n)*x^n = (1+x)/(1-mx+x^2) = o.g.f. of A_m, then Sum_{k=0..n} T(n,k)*a(k) = (m+2)^n. Related expansions of A_m are: A099493, A033999, A057078, A057077, A057079, A005408, A002878, A001834, A030221, A002315, A033890, A057080, A057081, A054320, A097783, A077416, A126866, A028230, A161591, for m=-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, respectively. - Philippe Deléham, Nov 16 2009
The Kn11, Kn12, Fi1 and Fi2 triangle sums link the triangle given above with three sequences; see the crossrefs. For the definitions of these triangle sums, see A180662. - Johannes W. Meijer, Apr 20 2011
4^n = (n-th row terms) dot (first n+1 odd integer terms). Example: 4^4 = 256 = (14, 28, 20, 7, 1) dot (1, 3, 5, 7, 9) = (14 + 84 + 100 + 49 + 9) = 256. - Gary W. Adamson, Jun 13 2011
The linear system of n equations with coefficients defined by the first n rows solve for diagonal lengths of regular polygons with N= 2n+1 edges; the constants c^0, c^1, c^2, ... are on the right hand side, where c = 2 + 2*cos(2*Pi/N). Example: take the first 4 rows relating to the 9-gon (nonagon), N = 2*4 + 1; with c = 2 + 2*cos(2*Pi/9) = 3.5320888.... The equations are (1,0,0,0) = 1; (1,1,0,0) = c; (2,3,1,0) = c^2; (5,9,5,1) = c^3. The solutions are 1, 2.53208..., 2.87938..., and 1.87938...; the four distinct diagonal lengths of the 9-gon (nonagon) with edge = 1. (Cf. comment in A089942 which uses the analogous operations but with c = 1 + 2*cos(2*Pi/9).) - Gary W. Adamson, Sep 21 2011
Also called the Lobb numbers, after Andrew Lobb, are a natural generalization of the Catalan numbers, given by L(m,n)=(2m+1)*Binomial(2n,m+n)/(m+n+1), where n >= m >= 0. For m=0, we get the n-th Catalan number. See added reference. - Jayanta Basu, Apr 30 2013
From Wolfdieter Lang, Sep 20 2013: (Start)
T(n, k) = A053121(2*n, 2*k). T(n, k) appears in the formula for the (2*n)-th power of the algebraic number rho(N):= 2*cos(Pi/N) = R(N, 2) in terms of the odd-indexed diagonal/side length ratios R(N, 2*k+1) = S(2*k, rho(N)) in the regular N-gon inscribed in the unit circle (length unit 1). S(n, x) are Chebyshev's S polynomials (see A049310):
  rho(N)^(2*n) = Sum_{k=0..n} T(n, k)*R(N, 2*k+1), n >= 0, identical in N > = 1. For a proof see the Sep 21 2013 comment under A053121. Note that this is the unreduced version if R(N, j) with j > delta(N), the degree of the algebraic number rho(N) (see A055034), appears.
  For the odd powers of rho(n) see A039598. (End)
Unsigned coefficients of polynomial numerators of Eqn. 2.1 of the Chakravarty and Kodama paper, defining the polynomials of A067311. - Tom Copeland, May 26 2016
The triangle is the Riordan square of the Catalan numbers in the sense of A321620. - Peter Luschny, Feb 14 2023

			Triangle T(n, k) begins:
  n\k     0     1     2     3     4     5    6   7   8  9
  0:      1
  1:      1     1
  2:      2     3     1
  3:      5     9     5     1
  4:     14    28    20     7     1
  5:     42    90    75    35     9     1
  6:    132   297   275   154    54    11    1
  7:    429  1001  1001   637   273    77   13   1
  8:   1430  3432  3640  2548  1260   440  104  15   1
  9:   4862 11934 13260  9996  5508  2244  663 135  17  1
  ... Reformatted by _Wolfdieter Lang_, Dec 21 2015
From _Paul Barry_, Feb 17 2011: (Start)
Production matrix begins
  1, 1,
  1, 2, 1,
  0, 1, 2, 1,
  0, 0, 1, 2, 1,
  0, 0, 0, 1, 2, 1,
  0, 0, 0, 0, 1, 2, 1,
  0, 0, 0, 0, 0, 1, 2, 1 (End)
From _Wolfdieter Lang_, Sep 20 2013: (Start)
Example for rho(N) = 2*cos(Pi/N) powers:
n=2: rho(N)^4 = 2*R(N,1) + 3*R(N,3) + 1*R(N, 5) =
  2 + 3*S(2, rho(N)) + 1*S(4, rho(N)), identical in N >= 1. For N=4 (the square with only one distinct diagonal), the degree delta(4) = 2, hence R(4, 3) and R(4, 5) can be reduced, namely to R(4, 1) = 1 and R(4, 5) = -R(4,1) = -1, respectively. Therefore, rho(4)^4 =(2*cos(Pi/4))^4 = 2 + 3 -1 = 4. (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
T. Myers and L. Shapiro, Some applications of the sequence 1, 5, 22, 93, 386, ... to Dyck paths and ordered trees, Congressus Numerant., 204 (2010), 93-104.

T. D. Noe, Rows n=0..50 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Quang T. Bach and Jeffrey B. Remmel, Generating functions for descents over permutations which avoid sets of consecutive patterns, arXiv:1510.04319 [math.CO], 2015 (see p.25).
M. Barnabei, F. Bonetti and M. Silimbani, Two permutation classes enumerated by the central binomial coefficients, arXiv preprint arXiv:1301.1790 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.3.8
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010), Article 10.8.2, example 15.
Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.
Paul Barry, Comparing two matrices of generalized moments defined by continued fraction expansions, arXiv preprint arXiv:1311.7161 [math.CO], 2013 and J. Int. Seq. 17 (2014) # 14.5.1.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Paul Barry, Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers, arXiv:2011.10827 [math.CO], 2020.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 15, 29.
Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 12.
Jonathan E. Beagley and Paul Drube, Combinatorics of Tableau Inversions, Electron. J. Combin., 22 (2015), #P2.44.
S. Chakravarty and Y. Kodama, A generating function for the N-soliton solutions of the Kadomtsev-Petviashvili II equation, arXiv preprint arXiv:0802.0524v2 [nlin.SI], 2008.
Wun-Seng Chou, Tian-Xiao He, and Peter J.-S. Shiue, On the Primality of the Generalized Fuss-Catalan Numbers, J. Int. Seqs., Vol. 21 (2018), #18.2.1.
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 11.
Paul Drube, Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers, arXiv:1606.04869 [math.CO], 2016.
Paul Drube, Generalized Path Pairs and Fuss-Catalan Triangles, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8.
T.-X. He and L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, example p 32.
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.
Thomas Koshy, Lobb's generalization of Catalan's parenthesization problem, The College Mathematics Journal 40 (2), March 2009, 99-107, DOI:10.1080/07468342.2009.11922344.
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 11.
Andrew Lobb, Deriving the n-th Catalan number, Mathematical Gazette, Vol. 83, No. 496 (March 1999), 109-110.
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174. See page 161.
Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, Sums of powers of Catalan triangle numbers, arXiv:1602.04347 [math.NT], 2016 (see 2.8).
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Wikipedia, Lobb number
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Columns give: A000108, A000245, A000344, A000588, A001392, A000589, A000590, A000012, A005408, A014107(n>1).
Row sums: A000984.
Triangle sums (see the comments): A000958 (Kn11), A001558 (Kn12), A088218 (Fi1, Fi2).
Cf. A008313, A039598, A084938, A000007, A067311, A321620.

/* As triangle */ [[Binomial(2*n, k+n)*(2*k+1)/(k+n+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 16 2015

T:=(n,k)->(2*k+1)*binomial(2*n,n-k)/(n+k+1): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form # Emeric Deutsch, May 06 2006
T := proc(n, k) option remember; if k = n then 1 elif k > n then 0 elif k = 0 then T(n-1, 0) + T(n-1,1) else T(n-1, k-1) + 2*T(n-1, k) + T(n-1, k+1) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..9) od; # Peter Luschny, Feb 14 2023

Table[Abs[Differences[Table[Binomial[2 n, n + i], {i, 0, n + 1}]]], {n, 0,7}] // Flatten (* Geoffrey Critzer, Dec 18 2011 *)
Join[{1},Flatten[Table[Binomial[2n-1,n-k]-Binomial[2n-1,n-k-2],{n,10},{k,0,n}]]] (* Harvey P. Dale, Dec 18 2011 *)
Flatten[Table[Binomial[2*n,m+n]*(2*m+1)/(m+n+1),{n,0,9},{m,0,n}]] (* Jayanta Basu, Apr 30 2013 *)

a(n, k) = (2*n+1)/(n+k+1)*binomial(2*k, n+k)
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(a(y, x), ", ")); print(""))
trianglerows(10) \\ Felix Fröhlich, Jun 24 2016

# Algorithm of L. Seidel (1877)
# Prints the first n rows of the triangle
def A039599_triangle(n) :
    D = [0]*(n+2); D[1] = 1
    b = True ; h = 1
    for i in range(2*n-1) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
            h += 1
        else :
            for k in range(1,h, 1) : D[k] += D[k+1]
        if b : print([D[z] for z in (1..h-1)])
        b = not b
A039599_triangle(10)  # Peter Luschny, May 01 2012

T(n,k) = C(2*n-1, n-k) - C(2*n-1, n-k-2), n >= 1, T(0,0) = 1.
From Emeric Deutsch, May 06 2006: (Start)
T(n,k) = (2*k+1)*binomial(2*n,n-k)/(n+k+1).
G.f.: G(t,z)=1/(1-(1+t)*z*C), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function. (End)
The following formulas were added by Philippe Deléham during 2003 to 2009: (Start)
Triangle T(n, k) read by rows; given by A000012 DELTA A000007, where DELTA is Deléham's operator defined in A084938.
T(n, k) = C(2*n, n-k)*(2*k+1)/(n+k+1). Sum(k>=0; T(n, k)*T(m, k) = A000108(n+m)); A000108: numbers of Catalan.
T(n, 0) = A000108(n); T(n, k) = 0 if k>n; for k>0, T(n, k) = Sum_{j=1..n} T(n-j, k-1)*A000108(j).
T(n, k) = A009766(n+k, n-k) = A033184(n+k+1, 2k+1).
G.f. for column k: Sum_{n>=0} T(n, k)*x^n = x^k*C(x)^(2*k+1) where C(x) = Sum_{n>=0} A000108(n)*x^n is g.f. for Catalan numbers, A000108.
T(0, 0) = 1, T(n, k) = 0 if n<0 or n=1, T(n, k) = T(n-1, k-1) + 2*T(n-1, k) + T(n-1, k+1).
a(n) + a(n+1) = 1 + A000108(m+1) if n = m*(m+3)/2; a(n) + a(n+1) = A039598(n) otherwise.
T(n, k) = A050165(n, n-k).
Sum_{j>=0} T(n-k, j)*A039598(k, j) = A028364(n, k).
Matrix inverse of the triangle T(n, k) = (-1)^(n+k)*binomial(n+k, 2*k) = (-1)^(n+k)*A085478(n, k).
Sum_{k=0..n} T(n, k)*x^k = A000108(n), A000984(n), A007854(n), A076035(n), A076036(n) for x = 0, 1, 2, 3, 4.
Sum_{k=0..n} (2*k+1)*T(n, k) = 4^n.
T(n, k)*(-2)^(n-k) = A114193(n, k).
Sum_{k>=h} T(n,k) = binomial(2n,n-h).
Sum_{k=0..n} T(n,k)*5^k = A127628(n).
Sum_{k=0..n} T(n,k)*7^k = A115970(n).
T(n,k) = Sum_{j=0..n-k} A106566(n+k,2*k+j).
Sum_{k=0..n} T(n,k)*6^k = A126694(n).
Sum_{k=0..n} T(n,k)*A000108(k) = A007852(n+1).
Sum_{k=0..floor(n/2)} T(n-k,k) = A000958(n+1).
Sum_{k=0..n} T(n,k)*(-1)^k = A000007(n).
Sum_{k=0..n} T(n,k)*(-2)^k = (-1)^n*A064310(n).
T(2*n,n) = A126596(n).
Sum_{k=0..n} T(n,k)*(-x)^k = A000007(n), A126983(n), A126984(n), A126982(n), A126986(n), A126987(n), A127017(n), A127016(n), A126985(n), A127053(n) for x=1,2,3,4,5,6,7,8,9,10 respectively.
Sum_{j>=0} T(n,j)*binomial(j,k) = A116395(n,k).
T(n,k) = Sum_{j>=0} A106566(n,j)*binomial(j,k).
T(n,k) = Sum_{j>=0} A127543(n,j)*A038207(j,k).
Sum_{k=0..floor(n/2)} T(n-k,k)*A000108(k) = A101490(n+1).
T(n,k) = A053121(2*n,2*k).
Sum_{k=0..n} T(n,k)*sin((2*k+1)*x) = sin(x)*(2*cos(x))^(2*n).
T(n,n-k) = Sum_{j>=0} (-1)^(n-j)*A094385(n,j)*binomial(j,k).
Sum_{j>=0} A110506(n,j)*binomial(j,k) = Sum_{j>=0} A110510(n,j)*A038207(j,k) = T(n,k)*2^(n-k).
Sum_{j>=0} A110518(n,j)*A027465(j,k) = Sum_{j>=0} A110519(n,j)*A038207(j,k) = T(n,k)*3^(n-k).
Sum_{k=0..n} T(n,k)*A001045(k) = A049027(n), for n>=1.
Sum_{k=0..n} T(n,k)*a(k) = (m+2)^n if Sum_{k>=0} a(k)*x^k = (1+x)/(x^2-m*x+1).
Sum_{k=0..n} T(n,k)*A040000(k) = A001700(n).
Sum_{k=0..n} T(n,k)*A122553(k) = A051924(n+1).
Sum_{k=0..n} T(n,k)*A123932(k) = A051944(n).
Sum_{k=0..n} T(n,k)*k^2 = A000531(n), for n>=1.
Sum_{k=0..n} T(n,k)*A000217(k) = A002457(n-1), for n>=1.
Sum{j>=0} binomial(n,j)*T(j,k)= A124733(n,k).
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000984(n), A089022(n), A035610(n), A130976(n), A130977(n), A130978(n), A130979(n), A130980(n), A131521(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
Sum_{k=0..n} T(n,k)*A005043(k) = A127632(n).
Sum_{k=0..n} T(n,k)*A132262(k) = A089022(n).
T(n,k) + T(n,k+1) = A039598(n,k).
T(n,k) = A128899(n,k)+A128899(n,k+1).
Sum_{k=0..n} T(n,k)*A015518(k) = A076025(n), for n>=1. Also Sum_{k=0..n} T(n,k)*A015521(k) = A076026(n), for n>=1.
Sum_{k=0..n} T(n,k)*(-1)^k*x^(n-k) = A033999(n), A000007(n), A064062(n), A110520(n), A132863(n), A132864(n), A132865(n), A132866(n), A132867(n), A132869(n), A132897(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
Sum_{k=0..n} T(n,k)*(-1)^(k+1)*A000045(k) = A109262(n), A000045:= Fibonacci numbers.
Sum_{k=0..n} T(n,k)*A000035(k)*A016116(k) = A143464(n).
Sum_{k=0..n} T(n,k)*A016116(k) = A101850(n).
Sum_{k=0..n} T(n,k)*A010684(k) = A100320(n).
Sum_{k=0..n} T(n,k)*A000034(k) = A029651(n).
Sum_{k=0..n} T(n,k)*A010686(k) = A144706(n).
Sum_{k=0..n} T(n,k)*A006130(k-1) = A143646(n), with A006130(-1)=0.
T(n,2*k)+T(n,2*k+1) = A118919(n,k).
Sum_{k=0..j} T(n,k) = A050157(n,j).
Sum_{k=0..2} T(n,k) = A026012(n); Sum_{k=0..3} T(n,k)=A026029(n).
Sum_{k=0..n} T(n,k)*A000045(k+2) = A026671(n).
Sum_{k=0..n} T(n,k)*A000045(k+1) = A026726(n).
Sum_{k=0..n} T(n,k)*A057078(k) = A000012(n).
Sum_{k=0..n} T(n,k)*A108411(k) = A155084(n).
Sum_{k=0..n} T(n,k)*A057077(k) = 2^n = A000079(n).
Sum_{k=0..n} T(n,k)*A057079(k) = 3^n = A000244(n).
Sum_{k=0..n} T(n,k)*(-1)^k*A011782(k) = A000957(n+1).
(End)
T(n,k) = Sum_{j=0..k} binomial(k+j,2j)*(-1)^(k-j)*A000108(n+j). - Paul Barry, Feb 17 2011
Sum_{k=0..n} T(n,k)*A071679(k+1) = A026674(n+1). - Philippe Deléham, Feb 01 2014
Sum_{k=0..n} T(n,k)*(2*k+1)^2 = (4*n+1)*binomial(2*n,n). - Werner Schulte, Jul 22 2015
Sum_{k=0..n} T(n,k)*(2*k+1)^3 = (6*n+1)*4^n. - Werner Schulte, Jul 22 2015
Sum_{k=0..n} (-1)^k*T(n,k)*(2*k+1)^(2*m) = 0 for 0 <= m < n (see also A160562). - Werner Schulte, Dec 03 2015
T(n,k) = GegenbauerC(n-k,-n+1,-1) - GegenbauerC(n-k-1,-n+1,-1). - Peter Luschny, May 13 2016
T(n,n-2) = A014107(n). - R. J. Mathar, Jan 30 2019
T(n,n-3) = n*(2*n-1)*(2*n-5)/3. - R. J. Mathar, Jan 30 2019
T(n,n-4) = n*(n-1)*(2*n-1)*(2*n-7)/6. - R. J. Mathar, Jan 30 2019
T(n,n-5) = n*(n-1)*(2*n-1)*(2*n-3)*(2*n-9)/30. - R. J. Mathar, Jan 30 2019

Corrected by Philippe Deléham, Nov 26 2009, Dec 14 2009

A357368 Triangle read by rows. Convolution triangle of the prime indicator sequence A089026.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 1, 10, 6, 1, 0, 5, 14, 21, 8, 1, 0, 1, 23, 47, 36, 10, 1, 0, 7, 28, 90, 108, 55, 12, 1, 0, 1, 49, 147, 258, 205, 78, 14, 1, 0, 1, 46, 249, 520, 595, 346, 105, 16, 1, 0, 1, 75, 360, 978, 1437, 1185, 539, 136, 18, 1

Offset: 0

Peter Luschny, Oct 03 2022

To clarify our terminology: We say T is the convolution triangle of a (or T is the partition transform of a), if a is a sequence of integers defined for n >= 1, and the transformation, as defined by the Maple function below, applied to a, returns T. In the generated lower triangular matrix T, i.e., in the convolution triangle of a, T(n, 1) = a(n) for n >= 1.
For instance, let a(n) = Bell(n), then we call A357583 the convolution triangle of the Bell numbers, but not A205574, which is also called the "Bell convolution triangle" in the comments but leads to a different triangle. Similarly, if a(n) = n!, then A090238 is the convolution triangle of the factorial numbers, not A084938. Third example: A128899 is the convolution triangle of the Catalan numbers in our setup, not A059365. More generally, we recommend that when computing the transform of a 0-based sequence to use only the terms for n >= 1 and not to shift the sequence.
Note that T is (0, 0)-based and the first column of a convolution triangle always is 1, 0, 0, 0, ... and the main diagonal is 1, 1, 1, 1, ... if a(1) = 1. The (1, 1)-based subtriangle of a genuine convolution triangle, i.e., a convolution triangle without column 0 and row 0, is often used in place of the convolution triangle but does not fit well into some applications of the convolution triangles.

			Triangle T(n, k) starts:
[0] 1;
[1] 0, 1;
[2] 0, 2,  1;
[3] 0, 3,  4,   1;
[4] 0, 1, 10,   6,   1;
[5] 0, 5, 14,  21,   8,   1;
[6] 0, 1, 23,  47,  36,  10,   1;
[7] 0, 7, 28,  90, 108,  55,  12,   1;
[8] 0, 1, 49, 147, 258, 205,  78,  14,  1;
[9] 0, 1, 46, 249, 520, 595, 346, 105, 16, 1;

Donald E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA]; Mathematica J. 2.1 (1992), no. 4, 67-78.
Peter Luschny, The P-transform.
Cormac O'Sullivan, De Moivre and Bell polynomials, arXiv:2203.02868 [math.CO], 2022.

Cf. A089026, A340991, A357588.

PMatrix := proc(dim, a) local n, k, m, g, M, A;
   if n = 0 then return [1] fi;
   A := [seq(a(i), i = 1..dim-1)];
   M := Matrix(dim, shape=triangular[lower]); M[1, 1] := 1;
   for m from 2 to dim do
       M[m, m] := M[m - 1, m - 1] * A[1];
       for k from m-1 by -1 to 2 do
           M[m, k] := add(A[i]*M[m-i, k-1], i = 1..m-k+1)
od od; M end:
a := n -> if isprime(n) then n else 1 fi: PMatrix(10, a);
# Alternatively, as the coefficients of row polynomials:
P := proc(n, x, a) option remember; ifelse(n = 0, 1,
    x*add(a(n - k)*P(k, x, a), k = 0..n-1)) end:
Pcoeffs := proc(n, a) seq(coeff(P(n, x, a), x, k), k=0..n) end:
seq(Pcoeffs(n, a), n = 0..9);
# Alternatively, term by term:
T := proc(n, k, a) option remember; # Alois P. Heinz style
    `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, a(n)),
    (q->add(T(j, q, a)*T(n-j, k-q, a), j=0..n))(iquo(k, 2)))) end:
seq(seq(T(n, k, a), k=0..n), n=0..9);

PMatrix[dim_, a_] := Module[{n, k, m, g, M, A}, If[n == 0, Return[1]]; A = Array[a, dim-1]; M = Array[0&, {dim, dim}]; M[[1, 1]] = 1; For[m = 2, m <= dim, m++, M[[m, m]] = M[[m-1, m-1]]*A[[1]]; For[k = m-1, k >= 2, k--, M[[m, k]] = Sum[A[[i]]*M[[m-i, k-1]], {i, 1, m-k+1}]]]; M];
a[n_] :=  If[PrimeQ[n], n, 1];
nmax = 10;
PM = PMatrix[nmax+1, a];
T[n_, k_] := PM[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 21 2022 *)

def ConvTriangle(dim: int, a) -> list[list[int]]:
    if callable(a): # Cache the input sequence.
        A = [a(i) for i in range(1, dim)]
    else:
        A = a
    print("In:", A)
    C = [[0 for k in range(m + 1)] for m in range(dim)]
    C[0][0] = 1
    for m in range(1, dim):
        C[m][m] = C[m - 1][m - 1] * A[0]
        for k in range(m - 1, 0, -1):
            C[m][k] = sum(A[i] * C[m - i - 1][k - 1] for i in range(m - k + 1))
    return C
from sympy import isprime, flatten
def a(n): return n if isprime(n) else 1
print(flatten(ConvTriangle(10, a)))

A275327 Triangle read by rows, Riordan array (1, (2+(x-1)/(2x^2)(1-sqrt(1-4x^2)))/ sqrt(1-4x^2)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 2, 7, 3, 1, 0, 10, 10, 12, 4, 1, 0, 5, 33, 25, 18, 5, 1, 0, 35, 42, 78, 48, 25, 6, 1, 0, 14, 144, 144, 155, 80, 33, 7, 1, 0, 126, 168, 420, 356, 275, 122, 42, 8, 1, 0, 42, 610, 723, 1018, 736, 450, 175, 52, 9, 1

Offset: 0

Peter Luschny, Aug 16 2016

			Table starts:
[n] [k=0,1,2,...] row sum
[0] [1] 1
[1] [0, 1] 1
[2] [0, 1, 1] 2
[3] [0, 3, 2, 1] 6
[4] [0, 2, 7, 3, 1] 13
[5] [0, 10, 10, 12, 4, 1] 37
[6] [0, 5, 33, 25, 18, 5, 1] 87
[7] [0, 35, 42, 78, 48, 25, 6, 1] 235
[8] [0, 14, 144, 144, 155, 80, 33, 7, 1] 578
[9] [0, 126, 168, 420, 356, 275, 122, 42, 8, 1] 1518

Cf. A057977 (column 1), A128899, A275328.

S := proc(n, k) option remember; local ecn:
if n = 0 then return n^k fi;
ecn := n -> n!/(iquo(n,2)!^2)/(iquo(n,2)+1);
add(ecn(i)*S(n-1,k-i), i=1..k-n+1) end:
A275327 := (n, k) -> S(k, n):
seq(seq(A275327(n, k),k=0..n),n=0..8);

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, (2+(#-1)/(2#^2) (1-Sqrt[1-4#^2]))/Sqrt[1-4#^2]&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

# uses[riordan_array from A256893]
s = (2+(x-1)/(2*x^2)*(1-sqrt(1-4*x^2)))/sqrt(1-4*x^2)
riordan_array(1, s, 12)

A370983 Triangle read by rows: T(n, k) = (n + k - 1)! / (k!*(n - k)!) if k > 0 and T(n, 0) = 0^n.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 3, 12, 20, 0, 4, 30, 120, 210, 0, 5, 60, 420, 1680, 3024, 0, 6, 105, 1120, 7560, 30240, 55440, 0, 7, 168, 2520, 25200, 166320, 665280, 1235520, 0, 8, 252, 5040, 69300, 665280, 4324320, 17297280, 32432400

Offset: 0

Peter Luschny, Mar 07 2024

A signed version of Catalan's triangle (version A128899) can be generated as the scaled inverse of this triangle. The scaled inverse of T is the inverse I of T post-processed by I(n, k) -> I(n, k)/I(n, n).

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 2,   3;
[3] 0, 3,  12,   20;
[4] 0, 4,  30,  120,   210;
[5] 0, 5,  60,  420,  1680,   3024;
[6] 0, 6, 105, 1120,  7560,  30240,  55440;
[7] 0, 7, 168, 2520, 25200, 166320, 665280, 1235520;

Cf. A006963 (main diagonal), A001813 (subdiagonal), A371028 (row sums).

Cf. A370706, A370707, A128899.

T := (n, k) -> `if`(k = 0, k^n, (n + k - 1)! / (k!*(n - k)!)):
seq(seq(T(n, k), k = 0..n), n = 0..9);
A370983 := (n, k) -> local j; ifelse(n = 0, 1, ifelse(k = 0, 0,
(-1)^k*mul((j - n) * (j + n) / (j + 1), j = 0..k - 1) / n)):

T[n_, k_] := If[n == 0, 1, If[k == 0, 0, (n + k - 1)! / (k! * (n - k)!)]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

from math import prod
def T(n, k):
    if n == 0: return 1
    if k == 0: return 0
    return (-1)**k * prod((j - n) * (j + n) / (j + 1) for j in range(k)) / n
for n in range(7): print([T(n, k) for k in range(n + 1)])

def A370983(n, k):
    if k  > n: return 0
    if n == 0: return 1
    if k == 0: return 0
    return binomial(n, k) * rising_factorial(n, k) // n
for n in range(7): print([A370983(n, k) for k in range(n + 1)])

# Added for the sake of reference only.
# For example ScaledInv(A370983, 7) gives the first seven rows of A128899.
def ScaledInv(T, dim): # We assume T(n, n) != 0 for all n.
    M = matrix(QQ, dim, T).inverse()
    for n in range(dim):
        c = M[n][n]
        M[n] = [M.row(n)[k] / c for k in range(dim)]
    return M

Assume n > 0 and k > 0 for the next four formulas:
T(n, k) = ((-1)^k / n) * Product_{j=0..k-1} ((j - n)*(j + n)/(j + 1)).
T(n, k) = binomial(n, k) * Pochhammer(n, k) / n.
T(n, k) = A370706(n, k) / n.
T(n, k) = A370707(n, k) / (n*k!).

A275326 Triangle read by rows, T(n,k) = ceiling(A275325(n,k)/2) for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 3, 0, 2, 1, 0, 10, 5, 0, 5, 4, 1, 0, 35, 28, 7, 0, 14, 14, 6, 1, 0, 126, 126, 54, 9, 0, 42, 48, 27, 8, 1, 0, 462, 528, 297, 88, 11, 0, 132, 165, 110, 44, 10, 1, 0, 1716, 2145, 1430, 572, 130, 13, 0, 429, 572, 429, 208, 65, 12, 1

Offset: 0

Peter Luschny, Aug 15 2016

An extension of the Catalan triangle A128899.

			Triangle starts:
[ n] [k=0,1,2,...] [row sum]
[ 0] [1] 1
[ 1] [0, 1] 1
[ 2] [0, 1] 1
[ 3] [0, 3] 3
[ 4] [0, 2, 1] 3
[ 5] [0, 10, 5] 15
[ 6] [0, 5, 4, 1] 10
[ 7] [0, 35, 28, 7] 70
[ 8] [0, 14, 14, 6, 1] 35
[ 9] [0, 126, 126, 54,  9] 315
[10] [0, 42, 48, 27, 8, 1] 126
[11] [0, 462, 528, 297, 88, 11] 1386
[12] [0, 132, 165, 110, 44, 10, 1] 462

Peter Luschny, Orbitals

Cf. A057977, A093178, A128899, A275324 (row sums), A275325.

# uses[orbital_factors]
# Function orbital_factors is in A275325.
def half_orbital_factors(n):
    F = orbital_factors(n)
    return [f//2 for f in F] if n >= 2 else F
for n in (0..12): print(half_orbital_factors(n))

T(n,k) = A275325(n,k)/2 for n>=2.
T(n,1) = A057977(n) for n>=1 (the extended Catalan numbers).
For odd n: T(n,1) = Sum_{k>=0} T(n+1,k).
Main diagonal: T(n, floor(n/2)) = A093178(n).

A337841 Triangle read by rows, T(n, k) = binomial(2n-1, 2k-1) * binomial(2n-2k, n-k) * (k+1) / binomial(n+k+1, n-k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 3, 3, 0, 6, 10, 4, 0, 14, 30, 21, 5, 0, 36, 90, 84, 36, 6, 0, 99, 275, 308, 180, 55, 7, 0, 286, 858, 1092, 780, 330, 78, 8, 0, 858, 2730, 3822, 3150, 1650, 546, 105, 9, 0, 2652, 8840, 13328, 12240, 7480, 3094, 840, 136, 10, 0, 8398, 29070, 46512, 46512, 31977, 15561, 5320, 1224, 171, 11

Offset: 0

Werner Schulte, Oct 30 2020

			The triangle T(n,k) for 0 <= k <= n starts:
  n\k:  0     1      2      3      4      5      6     7     8    9  10
=======================================================================
   0 :  1
   1 :  0     2
   2 :  0     3      3
   3 :  0     6     10      4
   4 :  0    14     30     21      5
   5 :  0    36     90     84     36      6
   6 :  0    99    275    308    180     55      7
   7 :  0   286    858   1092    780    330     78     8
   8 :  0   858   2730   3822   3150   1650    546   105     9
   9 :  0  2652   8840  13328  12240   7480   3094   840   136   10
  10 :  0  8398  29070  46512  46512  31977  15561  5320  1224  171  11
etc.

Cf. Row sums: A000984, main diagonal: A000027, 1st subdiagonal: A014105, 2nd subdiagonal: A055112, column 0: A000007, column 1: A007054.

T := proc(n, k) option remember; if k = n then n+1 else
T(n-1, k)*(2*n-2)*(2*n-1)/((n-1)*(n+2)-(k-1)*(k+2)) fi end:
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, Nov 02 2020

T(n,k) = binomial(2*n-1,n-k) * k * (2*k+1) * (2*k+2) / ((n+k)*(n+k+1)) for 1 <= k <= n, and T(n,0) = 0^n for n >= 0.
T(n,n) = n+1 for n >= 0; T(n,n-1) = (n-1) * (2*n-1) for n > 0; T(n,n-2) = (n-1) * (n-2) * (2*n-3) for n > 1.
T(n,k) = T(n-1,k) * (2*n-2) * (2*n-1) / ((n-1) * (n+2) - (k-1) * (k+2)) for 0 <= k < n with initial values T(n,n) = n+1 for n >= 0.
Row sums are A000984(n) for n >= 0.
Alternating row sums are 0 for n > 1.
Sum_{k=0..n} (-1)^k * T(n,k) * (k*(k+1)/2)^m = 0 for 0 <= m <= n-2.
T(n,1) = 12 * binomial(2*n-1,n-1)/((n+1)*(n+2)) = A007054(n) for n > 0.
T(n,k) = T(n,1)*(k*(k+1)*(2*k+1)/6)*binomial(n-1,k-1)/binomial(n+1+k,k-1) for 1 <= k <= n.
From Werner Schulte, Nov 09 2020: (Start)
T(n,k) = A128899(n,k) * (k+1) * (2*k+1) / (n+k+1) for 0 <= k <= n.
T(n,0) + Sum_{k=1..n} T(n,k) / (k*(k+1)) = A000108(n) for n >= 0. (End)

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A382394 a(n) = Sum_{k=0..n} A128899(n,k)^3.

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A359108 a(n) = A128899(2*n, n) = 2*binomial(4*n - 1, 3*n) for n >= 1 and a(0) = 1.

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A039599 Triangle formed from even-numbered columns of triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

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A357368 Triangle read by rows. Convolution triangle of the prime indicator sequence A089026.

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A275327 Triangle read by rows, Riordan array (1, (2+(x-1)/(2*x^2)*(1-sqrt(1-4*x^2)))/ sqrt(1-4*x^2)).

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A370983 Triangle read by rows: T(n, k) = (n + k - 1)! / (k!*(n - k)!) if k > 0 and T(n, 0) = 0^n.

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A275326 Triangle read by rows, T(n,k) = ceiling(A275325(n,k)/2) for n>=0 and 0<=k<=n.

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A359108 a(n) = A128899(2n, n) = 2binomial(4n - 1, 3n) for n >= 1 and a(0) = 1.

A275327 Triangle read by rows, Riordan array (1, (2+(x-1)/(2x^2)(1-sqrt(1-4x^2)))/ sqrt(1-4x^2)).

A337841 Triangle read by rows, T(n, k) = binomial(2n-1, 2k-1) * binomial(2n-2k, n-k) * (k+1) / binomial(n+k+1, n-k) for 0 <= k <= n.