A053122 -id:A053122 - cp's OEIS frontend

A030528 Triangle read by rows: a(n,k) = binomial(k,n-k).

1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 3, 4, 1, 0, 0, 1, 6, 5, 1, 0, 0, 0, 4, 10, 6, 1, 0, 0, 0, 1, 10, 15, 7, 1, 0, 0, 0, 0, 5, 20, 21, 8, 1, 0, 0, 0, 0, 1, 15, 35, 28, 9, 1, 0, 0, 0, 0, 0, 6, 35, 56, 36, 10, 1, 0, 0, 0, 0, 0, 1, 21, 70, 84, 45, 11, 1, 0, 0, 0, 0, 0, 0, 7, 56, 126, 120, 55, 12, 1

Offset: 1

Wolfdieter Lang

A convolution triangle of numbers obtained from A019590.
a(n,m) := s1(-1; n,m), a member of a sequence of triangles including s1(0; n,m)= A023531(n,m) (unit matrix) and s1(2; n,m)= A007318(n-1,m-1) (Pascal's triangle).
The signed triangular matrix a(n,m)*(-1)^(n-m) is the inverse matrix of the triangular Catalan convolution matrix A033184(n+1,m+1), n >= m >= 0, with A033184(n,m) := 0 if nRiordan array (1+x, x(1+x)). The signed triangle is the Riordan array (1-x,x(1-x)), inverse to (c(x),xc(x)) with c(x) g.f. for A000108. - Paul Barry, Feb 02 2005 [with offset 0]
Also, a(n,k)=number of compositions of n into k parts of 1's and 2's. Example: a(6,4)=6 because we have 2211, 2121, 2112, 1221, 1212 and 1122. - Emeric Deutsch, Apr 05 2005 [see MacMahon and Riordan. - Wolfdieter Lang, Jul 27 2023]
Subtriangle of A026729. - Philippe Deléham, Aug 31 2006
a(n,k) is the number of length n-1 binary sequences having no two consecutive 0's with exactly k-1 1's. Example: a(6,4)=6 because we have 01011, 01101, 01110, 10101, 10110, 11010. - Geoffrey Critzer, Jul 22 2013
Mirrored, shifted Fibonacci polynomials of A011973. The polynomials (illustrated below) of this entry have the property that p(n,t) = t * [p(n-1,t) + p(n-2,t)]. The additive properties of Pascal's triangle (A007318) are reflected in those of these polynomials, as can be seen in the Example Section below and also when the o.g.f. G(x,t) below is expanded as the series x*(1+x) + t * [x*(1+x)]^2 + t^2 * [x*(1+x)]^3 + ... . See also A053122 for a relation to Cartan matrices. - Tom Copeland, Nov 04 2014
Rows of this entry appear as columns of an array for an infinitesimal generator presented in the Copeland link. - Tom Copeland, Dec 23 2015
For n >= 2, the n-th row is also the coefficients of the vertex cover polynomial of the (n-1)-path graph P_{n-1}. - Eric W. Weisstein, Apr 10 2017
With an additional initial matrix element a_(0,0) = 1 and column of zeros a_(n,0) = 0 for n > 0, these are antidiagonals read from bottom to top of the numerical coefficients of the Maurer-Cartan form matrix of the Leibniz group L^(n)(1,1) presented on p. 9 of the Olver paper, which is generated as exp[c. * M] with (c.)^n = c_n and M the Lie infinitesimal generator A218272. Cf. A011973. And A169803. - Tom Copeland, Jul 02 2018

			Triangle starts:
  [ 1]  1
  [ 2]  1   1
  [ 3]  0   2   1
  [ 4]  0   1   3   1
  [ 5]  0   0   3   4   1
  [ 6]  0   0   1   6   5   1
  [ 7]  0   0   0   4  10   6   1
  [ 8]  0   0   0   1  10  15   7   1
  [ 9]  0   0   0   0   5  20  21   8   1
  [10]  0   0   0   0   1  15  35  28   9   1
  [11]  0   0   0   0   0   6  35  56  36  10   1
  [12]  0   0   0   0   0   1  21  70  84  45  11   1
  [13]  0   0   0   0   0   0   7  56 126 120  55  12   1
  ...
From _Tom Copeland_, Nov 04 2014: (Start)
For quick comparison to other polynomials:
  p(1,t) = 1
  p(2,t) = 1 + 1 t
  p(3,t) = 0 + 2 t + 1 t^2
  p(4,t) = 0 + 1 t + 3 t^2 + 1 t^3
  p(5,t) = 0 + 0   + 3 t^2 + 4 t^3 +  1 t^4
  p(6,t) = 0 + 0   + 1 t^2 + 6 t^3 +  5 t^4 +  1 t^5
  p(7,t) = 0 + 0   + 0     + 4 t^3 + 10 t^4 +  6 t^5 + 1 t^6
  p(8,t) = 0 + 0   + 0     + 1 t^3 + 10 t^4 + 15 t^5 + 7 t^6 + 1 t^7
  ...
Reading along columns gives rows for Pascal's triangle. (End)

P. A. MacMahon, Combinatory Analysis, Two volumes (bound as one), Chelsea Publishing Company, New York, 1960, Vol. I, nr. 124, p. 151.
John Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, London, 1958. eq. (35), p.124, 11. p. 154.

Indranil Ghosh, Rows 1..125 of triangle, flattened
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 18.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, Addendum to Elliptic Lie Triad
Loïc Foissy, Cointeraction on noncrossing partitions and related polynomial invariants, arXiv:2501.18212 [math.CO], 2025. See p. 21.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Donatella Merlini, Renzo Sprugnoli, and M. Cecilia Verri, An algebra for proper generating trees, Colloquium on Mathematics and Computer Science, Versailles, September 2000.
Donatella Merlini, Renzo Sprugnoli, and M. Cecilia Verri, An algebra for proper generating trees, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139.
Peter J. Olver, The canonical contact form.
Eric Weisstein's World of Mathematics, Path Graph
Eric Weisstein's World of Mathematics, Vertex Cover Polynomial

Row sums A000045(n+1) (Fibonacci). a(n, 1)= A019590(n) (Fermat's last theorem). Cf. A049403.

Cf. A104597, A146559, A155020, A125145, A057078, A039717, A001792, A057862, A011973, A115139.

/* As triangle */ [[Binomial(k, n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Nov 05 2014

for n from 1 to 12 do seq(binomial(k,n-k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Apr 05 2005

nn=10;CoefficientList[Series[(1+x)/(1-y x - y x^2),{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Jul 22 2013 *)
Table[Binomial[k, n - k], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Dec 23 2015 *)
CoefficientList[Table[x^(n/2 - 1) Fibonacci[n + 1, Sqrt[x]], {n, 10}],
   x] // Flatten (* Eric W. Weisstein, Apr 10 2017 *)

a(n, m) = 2*(2*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

G.f. for m-th column: (x*(1+x))^m.

As a number triangle with offset 0, this is T(n, k) = Sum_{i=0..n} (-1)^(n+i)*binomial(n, i)*binomial(i+k+1, 2k+1). The antidiagonal sums give the Padovan sequence A000931(n+5). Inverse binomial transform of A078812 (product of lower triangular matrices). - Paul Barry, Jun 21 2004

G.f.: (1 + x)/(1 - y*x - y*x^2). - Geoffrey Critzer, Jul 22 2013 [offset 0] [with offset 1: g.f. of row polynomials in y: x*(1+x)*y/(1 - x*(1+x)*y). - Wolfdieter Lang, Jul 27 2023]

From Tom Copeland, Nov 04 2014: (Start)

O.g.f: G(x,t) = x*(1+x) / [1 - t*x*(1+x)] = -P[Cinv(-x),t], where P(x,t)= x / (1 + t*x) and Cinv(x)= x*(1-x) are the compositional inverses in x of Pinv(x,t) = -P(-x,t) = x / (1 - t*x) and C(x) = [1-sqrt(1-4*x)]/2, an o.g.f. for the shifted Catalan numbers A000108.

Therefore, Ginv(x,t) = -C[Pinv(-x,t)] = {-1 + sqrt[1 + 4*x/(1+t*x)]}/2, which is -A124644(-x,t).

This places this array in a family of arrays related by composition of P and C and their inverses and interpolation by t, such as A091867 and A104597, and associated to the Catalan, Motzkin, Fine, and Fibonacci numbers. Cf. A104597 (polynomials shifted in t) A125145, A146559, A057078, A000045, A155020, A125145, A039717, A001792, A057862, A011973, A115139. (End)

More terms from Emeric Deutsch, Apr 05 2005

A078812 Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 10, 6, 1, 5, 20, 21, 8, 1, 6, 35, 56, 36, 10, 1, 7, 56, 126, 120, 55, 12, 1, 8, 84, 252, 330, 220, 78, 14, 1, 9, 120, 462, 792, 715, 364, 105, 16, 1, 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1, 11, 220, 1287, 3432, 5005, 4368, 2380, 816, 171, 20, 1

Offset: 0

Michael Somos, Dec 05 2002

Warning: formulas and programs sometimes refer to offset 0 and sometimes to offset 1.
Apart from signs, identical to A053122.
Coefficient array for Morgan-Voyce polynomial B(n,x); see A085478 for references. - Philippe Deléham, Feb 16 2004
T(n,k) is the number of compositions of n having k parts when there are q kinds of part q (q=1,2,...). Example: T(4,2) = 10 because we have (1,3),(1,3'),(1,3"), (3,1),(3',1),(3",1),(2,2),(2,2'),(2',2) and (2',2'). - Emeric Deutsch, Apr 09 2005
T(n, k) is also the number of idempotent order-preserving full transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|). - Abdullahi Umar, Oct 02 2008
This sequence is jointly generated with A085478 as a triangular array of coefficients of polynomials v(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = u(n-1,x) + x*v(n-1)x and v(n,x) = u(n-1,x) + (x+1)*v(n-1,x). See the Mathematica section. - Clark Kimberling, Feb 25 2012
Concerning Kimberling's recursion relations, see A102426. - Tom Copeland, Jan 19 2016
Subtriangle of the triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 27 2012
From Wolfdieter Lang, Aug 30 2012: (Start)
With offset [0,0] the triangle with entries R(n,k) = T(n+1,k+1):= binomial(n+k+1, 2*k+1), n >= k >= 0, and zero otherwise, becomes the Riordan lower triangular convolution matrix R = (G(x)/x, G(x)) with G(x):=x/(1-x)^2 (o.g.f. of A000027). This means that the o.g.f. of column number k of R is (G(x)^(k+1))/x. This matrix R is the inverse of the signed Riordan lower triangular matrix A039598, called in a comment there S.
The Riordan matrix with entries R(n,k), just defined, provides the transition matrix between the sequence entry F(4*m*(n+1))/L(2*l), with m >= 0, for n=0,1,... and the sequence entries 5^k*F(2*m)^(2*k+1) for k = 0,1,...,n, with F=A000045 (Fibonacci) and L=A000032 (Lucas). Proof: from the inverse of the signed triangle Riordan matrix S used in a comment on A039598.
For the transition matrix R (T with offset [0,0]) defined above, row n=2: F(12*m) /L(2*m) = 3*5^0*F(2*m)^1 + 4*5^1*F(2*m)^3 + 1*5^2*F(2*m)^5, m >= 0. (End)
From R. Bagula's comment in A053122 (cf. Damianou link p. 10), this array gives the coefficients (mod sign) of the characteristic polynomials for the Cartan matrix of the root system A_n. - Tom Copeland, Oct 11 2014
For 1 <= k <= n, T(n,k) equals the number of (n-1)-length ternary words containing k-1 letters equal 2 and avoiding 01. - Milan Janjic, Dec 20 2016
The infinite sum (Sum_{i >= 0} (T(s+i,1+i) / 2^(s+2*i)) * zeta(s+1+2*i)) = 1 allows any zeta(s+1) to be expressed as a sum of rational multiples of zeta(s+1+2*i) having higher arguments. For example, zeta(3) can be expressed as a sum involving zeta(5), zeta(7), etc. The summation for each s >= 1 uses the s-th diagonal of the triangle. - Robert B Fowler, Feb 23 2022
The convolution triangle of the nonnegative integers. - Peter Luschny, Oct 07 2022

			Triangle begins, 1 <= k <= n:
                          1
                        2   1
                      3   4   1
                    4  10   6   1
                  5  20  21   8   1
                6  35  56  36  10   1
              7  56 126 120  55  12   1
            8  84 252 330 220  78  14   1
From _Peter Bala_, Feb 11 2025: (Start)
The array factorizes as an infinite product of lower triangular arrays:
  / 1               \    / 1              \ / 1              \ / 1             \
  | 2    1           |   | 2   1          | | 0  1           | | 0  1          |
  | 3    4   1       | = | 3   2   1      | | 0  2   1       | | 0  0  1       | ...
  | 4   10   6   1   |   | 4   3   2  1   | | 0  3   2  1    | | 0  0  2  1    |
  | 5   20  21   8  1|   | 5   4   3  2  1| | 0  4   3  2  1 | | 0  0  3  2  1 |
  |...               |   |...             | |...             | |...            |
Cf. A092276. (End)

T. D. Noe, Rows n = 0..50 of triangle, flattened
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - From _N. J. A. Sloane_, May 10 2012
Peter Bala, Factorisations of some Riordan arrays as infinite products
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers, arXiv:2011.10827 [math.CO], 2020.
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
Nour-Eddine Fahssi, On the combinatorics of exclusion in Haldane fractional statistics, arXiv:1808.00045 [cond-mat.stat-mech], 2018.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
A. Laradji, and A. Umar, Combinatorial results for semigroups of order-preserving full transformations, Semigroup Forum 72 (2006), 51-62.
Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart. 43, no. 4, (2005) 359-370.

This triangle is formed from odd-numbered rows of triangle A011973 read in reverse order.
Row sums give A001906. With signs: A053122.
The column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for k=1..6, resp. For k=7..24 they are A010966..(+2)..A011000 and for k=25..50 they are A017713..(+2)..A017763.
Cf. A128908, A053123, A049310, A119900, A085478, A029653, A106195.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n+k+1, 2*k+1) ))); # G. C. Greubel, Aug 01 2019

a078812 n k = a078812_tabl !! n !! k
a078812_row n = a078812_tabl !! n
a078812_tabl = [1] : [2, 1] : f [1] [2, 1] where
   f us vs = ws : f vs ws where
     ws = zipWith (-) (zipWith (+) ([0] ++ vs) (map (* 2) vs ++ [0]))
                      (us ++ [0, 0])
-- Reinhard Zumkeller, Dec 16 2013

/* As triangle */ [[Binomial(n+k-1, 2*k-1): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jun 01 2018

for n from 1 to 11 do seq(binomial(n+k-1,2*k-1),k=1..n) od; # yields sequence in triangular form; Emeric Deutsch, Apr 09 2005
# Uses function PMatrix from A357368. Adds a row and column above and to the left.
PMatrix(10, n -> n); # Peter Luschny, Oct 07 2022

(* First program *)
u[1, x_]:= 1; v[1, x_]:= 1; z = 13;
u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
v[n_, x_]:= u[n-1, x] + (x+1)*v[n-1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A085478 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A078812 *) (* Clark Kimberling, Feb 25 2012 *)
(* Second program *)
Table[Binomial[n+k+1, 2*k+1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 01 2019 *)

T(n,m):=sum(binomial(2*k,n-m)*binomial(m+k,k)*(-1)^(n-m+k)*binomial(n+1,m+k+1),k,0,n-m); /* Vladimir Kruchinin, Apr 13 2016 */

{T(n, k) = if( n<0, 0, binomial(n+k-1, 2*k-1))};

{T(n, k) = polcoeff( polcoeff( x*y / (1 - (2 + y) * x + x^2) + x * O(x^n), n), k)};

@cached_function
def T(k,n):
    if k==n: return 1
    if k==0: return 0
    return sum(i*T(k-1,n-i) for i in (1..n-k+1))
A078812 = lambda n,k: T(k,n)
[[A078812(n,k) for k in (1..n)] for n in (1..8)] # Peter Luschny, Mar 12 2016

[[binomial(n+k+1, 2*k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

G.f.: x*y / (1 - (2 + y)*x + x^2). To get row n, expand this in powers of x then expand the coefficient of x^n in increasing powers of y.
From Philippe Deléham, Feb 16 2004: (Start)
If indexing begins at 0 we have
T(n,k) = (n+k+1)!/((n-k)!*(2k+1))!.
T(n,k) = Sum_{j>=0} T(n-1-j, k-1)*(j+1) with T(n, 0) = n+1, T(n, k) = 0 if n < k.
T(n,k) = T(n-1, k-1) + T(n-1, k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j) with T(n,k) = 0 if k < 0, T(0, 0)=1 and T(0, k) = 0 for k > 0.
G.f. for the column k: Sum_{n>=0} T(n, k)*x^n = (x^k)/(1-x)^(2k+2).
Row sums: Sum_{k>=0} T(n, k) = A001906(n+1). (End)
Antidiagonal sums are A000079(n) = Sum_{k=0..floor(n/2)} binomial(n+k+1, n-k). - Paul Barry, Jun 21 2004
Riordan array (1/(1-x)^2, x/(1-x)^2). - Paul Barry, Oct 22 2006
T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n, T(n,k) = T(n-1,k-1) + 2*T(n-1,k) - T(n-2,k). - Philippe Deléham, Jan 26 2010
For another version see A128908. - Philippe Deléham, Mar 27 2012
T(n,m) = Sum_{k=0..n-m} (binomial(2*k,n-m)*binomial(m+k,k)*(-1)^(n-m+k)* binomial(n+1,m+k+1)). - Vladimir Kruchinin, Apr 13 2016
T(n, k) = T(n-1, k) + (T(n-1, k-1) + T(n-2, k-1) + T(n-3, k-1) + ...) for k >= 2 with T(n, 1) = n. - Peter Bala, Feb 11 2025
From Peter Bala, May 04 2025: (Start)
With the column offset starting at 0, the n-th row polynomial B(n, x) = 1/sqrt(x + 4) * Chebyshev_U(2*n+1, (1/2)*sqrt(x + 4)) = (-1)^n * Chebyshev_U(n, -(1/2)*(x + 2)).
B(n, x) / Product_{k = 1..2*n} (1 + 1/B(k, x)) = b(n, x), the n-th row polynomial of A085478. (End)

Edited by N. J. A. Sloane, Apr 28 2008

A034867 Triangle of odd-numbered terms in rows of Pascal's triangle.

Original entry on oeis.org

1, 2, 3, 1, 4, 4, 5, 10, 1, 6, 20, 6, 7, 35, 21, 1, 8, 56, 56, 8, 9, 84, 126, 36, 1, 10, 120, 252, 120, 10, 11, 165, 462, 330, 55, 1, 12, 220, 792, 792, 220, 12, 13, 286, 1287, 1716, 715, 78, 1, 14, 364, 2002, 3432, 2002, 364, 14, 15, 455, 3003, 6435, 5005, 1365, 105, 1

Offset: 0

N. J. A. Sloane

Also triangle of numbers of n-sequences of 0,1 with k subsequences of consecutive 01 because this number is C(n+1,2*k+1). - Roger Cuculiere (cuculier(AT)imaginet.fr), Nov 16 2002
From Gary W. Adamson, Oct 17 2008: (Start)
Received from Herb Conn:
Let T = tan x, then
  tan x = T
  tan 2x = 2T / (1 - T^2)
  tan 3x = (3T - T^3) / (1 - 3T^2)
  tan 4x = (4T - 4T^3) / (1 - 6T^2 + T^4)
  tan 5x = (5T - 10T^3 + T^5) / (1 - 10T^2 + 5T^4)
  tan 6x = (6T - 20T^3 + 6T^5) / (1 - 15T^2 + 15T^4 - T^6)
  tan 7x = (7T - 35T^3 + 21T^5 - T^7) / (1 - 21T^2 + 35T^4 - 7T^6)
  tan 8x = (8T - 56T^3 + 56T^5 - 8T^7) / (1 - 28T^2 + 70T^4 - 28T^6 + T^8)
  tan 9x = (9T - 84T^3 + 126T^5 - 36T^7 + T^9) / (1 - 36 T^2 + 126T^4 - 84T^6 + 9T^8)
... To get the next one in the series, (tan 10x), for the numerator add:
  9....84....126....36....1 previous numerator +
  1....36....126....84....9 previous denominator =
  10..120....252...120...10 = new numerator
For the denominator add:
  ......9.....84...126...36...1 = previous numerator +
  1....36....126....84....9.... = previous denominator =
  1....45....210...210...45...1 = new denominator
...where numerators = A034867, denominators = A034839
(End)
Column k is the sum of columns 2k and 2k+1 of A007318. - Philippe Deléham, Nov 12 2008
Triangle, with zeros omitted, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011
The row polynomials N(n,x) = Sum_{k=0..floor((n-1)/2)} T(n-1,k)*x^k, and D(n,x) = Sum_{k=0..floor(n/2)} A034839(n,k)*x^k, n >= 1, satisfy the recurrences N(n,x) = D(n-1,x) + N(n-1,x), D(n,x) = D(n-1,x) + x*N(n-1,x), with inputs N(1,x) = 1 = D(1,x). This is due to the Pascal triangle A007318 recurrence. Q(n,x) := tan(n*x)/tan(x) satisfies the recurrence Q(n,x) = (1 + Q(n-1,x))/(1 - v(x)*Q(n-1,x)) with input Q(1,x) = 1 and v = v(x) := (tan(x))^2. This recurrence is obtained from the addition theorem for tan(n*x) using n = 1 + (n-1). Therefore Q(n,x) = N(n,-v(x))/D(n,-v(x)). This proves the Gary W. Adamson contribution from above. See also A220673. This calculation was motivated by an e-mail of Thomas Olsen. The Oliver/Prodinger and Ma references resort to HAKEM Al Memo 239, Item 16, for the tan(n*x) formula in terms of tan(x). - Wolfdieter Lang, Jan 17 2013
The infinitesimal generator (infinigen) for the Narayana polynomials A090181/A001263 can be formed from the row polynomials P(n,y) of this entry. The resulting matrix is an instance of a matrix representation of the analytic infinigens presented in A145271 for general sets of binomial Sheffer polynomials and in A001263 and A119900 specifically for the Narayana polynomials. Given the column vector of row polynomials V = (1, P(1,x) = 2x, P(2,y) = 3x + x^2, P(3,y) = 4x + 4x^2, ...), form the lower triangular matrix M(n,k) = V(n-k,n-k), i.e., diagonally multiply the matrix with all ones on the diagonal and below by the components of V. Form the matrix MD by multiplying A132440^Transpose = A218272 = D (representing derivation of o.g.f.s) by M, i.e., MD = M*D. The non-vanishing component of the first row of (MD)^n * V / (n+1)! is the n-th Narayana polynomial. - Tom Copeland, Dec 09 2015
The diagonals of this entry are A078812 (also shifted A128908 and unsigned A053122, which are embedded in A030528, A102426, A098925, A109466, A092865). Equivalently, the antidiagonals of A078812 are the rows of A034867. - Tom Copeland, Dec 12 2015
Binomial(n,2k+1) is also the number of permutations avoiding both 132 and 213 with k peaks, i.e., positions with w[i]w[i+2]. - Lara Pudwell, Dec 19 2018
Binomial(n,2k+1) is also the number of permutations avoiding both 123 and 132 with k peaks, i.e., positions with w[i]w[i+2]. - Lara Pudwell, Dec 19 2018
The row polynomial P(n, x) = Sum_{0..floor(n/2)} T(n, k)*x^k appears as numerator polynomial of the diagonal sequence m of triangle A104698 as follows. G(m, x) = P(m, x^2)/(1 - x)^(m+1), for m >= 0. - Wolfdieter Lang, May 14 2025
Number of acyclic orientations of the path graph on n+1 vertices, with k-1 sinks. - Per W. Alexandersson, Aug 15 2025

			Triangle T starts:
  n\k   0   1   2   3   4  5 ...   ----------------------------------------
0:    1
1:    2
2:    3   1
3:    4   4
4:    5  10   1
5:    6  20   6
6:    7  35  21   1
7:    8  56  56   8
8:    9  84 126  36   1
9:   10 120 252 120  10
 10:   11 165 462 330  55  1
 11:   12 220 792 792 220 12
... ... reformatted and extended by - _Wolfdieter Lang_, May 14 2025

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 136.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.
L. Carlitz and R. Scoville, Zero-one sequences and Fibonacci numbers, Fibonacci Quarterly, 15 (1977), 246-254.
Sergi Elizalde, Johnny Rivera Jr., and Yan Zhuang, Counting pattern-avoiding permutations by big descents, arXiv:2408.15111 [math.CO], 2024. See p. 6.
S.-M. Ma, On some binomial coefficients related to the evaluation of tan(nx), arXiv preprint arXiv:1205.0735 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 13 2012
K. Oliver and H. Prodinger, The continued fraction expansion of Gauss' hypergeometric function and a new application to the tangent function, Transactions of the Royal Society of South Africa, Vol. 76 (2012), 151-154, [DOI], [PDF]. - From _N. J. A. Sloane_, Jan 03 2013
Eric Weisstein's World of Mathematics, Tangent. [From _Eric W. Weisstein_, Oct 18 2008]

Cf. A000129, A007318, A034839, A034867, A084938, A131980, A220673.
Cf. A001263, A090181, A119900, A132440, A145271.
Cf. A030528, A053122, A078812, A092865, A098925, A102426, A109466, A128908, A218272, A104698.
From Wolfdieter Lang, May 14 2025:(Start)
Row length A008619. Row sums A000079. Alternating row sums  A009545(n+1).
Column sequences (with certain offsets): A000027, A000292, A000389, A000580, A000582, A001288, ... (End)

/* as a triangle */ [[Binomial(n+1,2*k+1): k in [0..Floor(n/2)]]: n in [0..20]]; // G. C. Greubel, Mar 06 2018

seq(seq(binomial(n+1,2*k+1), k=0..floor(n/2)), n=0..14); # Emeric Deutsch, Apr 01 2005

u[1, x_] := 1; v[1, x_] := 1; z = 12;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]  (* A034839 as a triangle *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]  (* A034867 as a triangle *)
(* Clark Kimberling, Feb 18 2012 *)
Table[Binomial[n+1, 2*k+1], {n,0,20}, {k,0,Floor[n/2]}]//Flatten (* G. C. Greubel, Mar 06 2018 *)

for(n=0,20, for(k=0,floor(n/2), print1(binomial(n+1,2*k+1), ", "))) \\ G. C. Greubel, Mar 06 2018

T(n,k) = C(n+1,2k+1) = Sum_{i=k..n-k} C(i,k) * C(n-i,k).
E.g.f.: 1+(exp(x)*sinh(x*sqrt(y)))/sqrt(y). - Vladeta Jovovic, Mar 20 2005
G.f.: 1/((1-z)^2-t*z^2). - Emeric Deutsch, Apr 01 2005
T(n,k) = Sum_{j = 0..n} A034839(j,k). - Philippe Deléham, May 18 2005
Pell(n+1) = A000129(n+1) = Sum_{k=0..n} T(n,k) * 2^k = (1/n!) Sum_{k=0..n} A131980(n,k) * 2^k. - Tom Copeland, Nov 30 2007
T(n,k) = A007318(n,2k) + A007318(n,2k+1). - Philippe Deléham, Nov 12 2008
O.g.f for column k, k>=0: (1/(1-x)^2)*(x/(1-x))^(2*k). See the G.f. of this array given above by Emeric Deutsch. - Wolfdieter Lang, Jan 18 2013
T(n,k) = (x^(2*k+1))*((1+x)^n-(1-x)^n)/2. - L. Edson Jeffery, Jan 15 2014

More terms from Emeric Deutsch, Apr 01 2005

A115139 Array of coefficients of polynomials related to integer powers of the generating function of Catalan numbers A000108.

Original entry on oeis.org

1, 1, 1, -1, 1, -2, 1, -3, 1, 1, -4, 3, 1, -5, 6, -1, 1, -6, 10, -4, 1, -7, 15, -10, 1, 1, -8, 21, -20, 5, 1, -9, 28, -35, 15, -1, 1, -10, 36, -56, 35, -6, 1, -11, 45, -84, 70, -21, 1, 1, -12, 55, -120, 126, -56, 7, 1, -13, 66, -165, 210, -126, 28, -1, 1, -14, 78, -220, 330, -252, 84, -8, 1, -15, 91, -286, 495, -462, 210, -36, 1

Offset: 1

Wolfdieter Lang, Jan 13 2006

This is a signed version of A011973 (Fibonacci polynomials) with different offset.
The sequence of row lengths is [1,1,2,2,3,3,4,4,5,5,6,6,...] = A008619(n-1), n>=1.
The row sums give the period 6 sequence [1,1,0,-1,-1,0,...] = A010892(n-1), n>=1.
The o.g.f. for the column m sequence (with leading zeros) is ((-1)^m)*x^(2*m+1)/(1-x)^(m+1).
The unsigned row sums give the Fibonacci numbers A000045(n-1), n>=1.
The row polynomial are P(n,x):= Sum_{m=0..ceiling(n/2)-1} a(n,m)*x^m = (sqrt(x)^(n-1))*S(n-1,1/sqrt(x)), n>=1, with Chebyshev's S(n,x) polynomials A049310.
These polynomials appear in the formula 1/c(x)^n = P(n+1,x) - x*P(n,x)*c(x), n>=1, with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers). See the W. Lang reference, eqs. (1) and (2), p. 408, with P(n,x):=p(-n,x).
These polynomials also appear in the formula c(x)^n = (-P(n-1,x) + P(n,x)*c(x))/x^(n-1), n>=1, with the above given o.g.f. c(x) of A000108 (Catalan numbers). See the W. Lang reference, eq. (1), with P(n,x):=p(-n,x).
With offset n>=0 this array a(n,m) coincides with the row reversed coefficient table of Chebyshev's S-polynomials without interspersed zeros. See A049310 for the S(n,x) coefficient table with increasing powers of x.
The polynomials with this sequence as coefficients form the set of so-called "Catalan polynomials", having arisen from computations in looking at the problem of 'fitting' iterated generating function schemes to the Catalan sequence. A neighboring pair forms the basis of a first-order linear recurrence that generates, through a succession of iterated generating functions (polynomials in Z[x]), a predetermined number of Catalan numbers before 'failing' - see the Clapperton et al. 2008 reference in Utilitas Mathematica, where some of the essential mathematical properties of the Catalan polynomials are also listed (based mainly on existing results for Dickson and Chebyshev polynomials, to which they are related). - Peter J Larcombe, Sep 16 2008
In the Clapperton et al. 2008 Congressus Numerantium paper, a new class of nonlinear identities satisfied by Catalan polynomials are presented. They arise from the algebraic implementation of particular cases of a general root finding formulation due to Householder, of which the classic O(2) Newton-Raphson and O(3) Halley algorithms are special cases. The role of Catalan polynomials in forming Padé approximants to the Catalan sequence o.g.f. is also discussed. - Peter J Larcombe, Nov 02 2008
These polynomials appear in the following statements: (i) P(k+1,x)/P(k+2,x) is the g.f. of all ordered trees (Dyck paths) of height at most k; (ii) x^k/(P(k+1,x)*P(k+2,x)) is the g.f. of all ordered trees (Dyck paths) of height k. See the de Bruijn et al., the Kreweras, the Sedgewick and Flajolet (p. 258), and the Flajolet and Sedgewick (p. 326) references. - Emeric Deutsch, Jun 16 2011
For a mirrored, shifted version showing the relation of these coefficients to the Pascal triangle, Fibonacci, and other number triangles, see A030528. See also A053122 for a relation to Cartan matrices. (Cf. A011973, A169803, A115139, A092865, A098925, and A102426.) - Tom Copeland, Nov 04 2014
M. Sinan Kul, Dec 09 2015, observed that (in a rewritten form) Chebyshev's S polynomials A049310 are given by S(n, x) = Sum_{m=0..floor(n/2)} a(n+1, m)*x^(n-2*m), n >= 0. This formula is well known and can be proved from the S recurrence by induction using the recurrence for the binomial coefficients. - Wolfdieter Lang, Feb 01 2016
These are the coefficients of generalized Fibonacci polynomials (see link bellow). - Rigoberto Florez, Aug 28 2022

			The irregular triangle a(n, m) begins:
  n\m  0   1   2    3   4    5   6   7  8
  1:   1
  2:   1
  3:   1  -1
  4:   1  -2
  5:   1  -3   1
  6:   1  -4   3
  7:   1  -5   6   -1
  8:   1  -6  10   -4
  9:   1  -7  15  -10   1
  10:  1  -8  21  -20   5
  11:  1  -9  28  -35  15   -1
  12:  1 -10  36  -56  35   -6
  13:  1 -11  45  -84  70  -21   1
  14:  1 -12  55 -120 126  -56   7
  15:  1 -13  66 -165 210 -126  28  -1
  16:  1 -14  78 -220 330 -252  84  -8
  17:  1 -15  91 -286 495 -462 210 -36  1
  ... Reformatted and extended. - _Wolfdieter Lang_, Jan 27 2016
1/c(x) = P(2,x) - x*P(1,x)*c(x) = 1 - x*c(x), with the o.g.f. of A000108 (Catalan).
1/c(x)^2 = P(3,x) - x*P(2,x)*c(x) = (1-x) - x*c(x).
c(x)^2 = (-P(1,x) + P(2,x)*c(x))/x^1 = (-1 + 1*c(x))/x.
c(x)^3 = (-P(2,x) + P(3,x)*c(x))/x^2 = (-1 + (1-x)*c(x))/x^2.
P(3,x) = 1-x = x*S(2,1/sqrt(x)) with Chebyshev's S(2,y) = U(2,y/2) = y^2 - 1.

J. A. Clapperton, P. J. Larcombe and E. J. Fennessey, On iterated generating functions for integer sequences and Catalan polynomials, Utilitas Mathematica, 77 (2008), 3-33.
J. A. Clapperton, P. J. Larcombe, E. J. Fennessey and P. Levrie, A class of auto-identities for Catalan polynomials and Padé approximation, Congressus Numerantium, 189 (2008), 77-95.
N. G. de Bruijn, D. E. Knuth and S. O. Rice, The average height of planted plane trees, in: Graph Theory and Computing (ed. T. C. Read), Academic Press, New York, 1972, pp. 15-22.
R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996.

Alois P. Heinz, Rows n = 1..200, flattened
Hong Duc Bui, Counting Number of Triangulations of Point Sets: Reinterpreting and Generalizing the Triangulation Polynomials, arXiv:2504.09615 [cs.CG], 2025. See p. 3.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009.
R. Florez and J. C. Saunders, Irreducibility of generalized Fibonacci polynomials , Integers 22 (2022).
Atsushi Komaba, Hisashi Johno, and Kazunori Nakamoto, A novel statistical approach for two-sample testing based on the overlap coefficient, arXiv:2206.03166 [math.ST], 2022.
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
Wolfdieter Lang, First 16 rows
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419.
Peter J. Larcombe and Eric J. Fennessey, A Non-Linear Identity for A Particular Class of Polynomial Families, Fibonacci Quart. 52 (2014), no. 1, 75-79. Mentions this sequence.

Cf. A011973, A030528, A053122, A092865, A098925, A102426, A115139, A169803.

seq(seq((-1)^k*binomial(n-k,k),k=0..floor(n/2)),n=0..16); # Peter Luschny, May 10 2016

p[x_, n_] := p[x, n] = p[x, n - 1] + x*p[x, n - 2];
p[x_, -1] = p[x_, 0] = 1; p[x_, 1] = 1 + x;
Flatten[ Table[ CoefficientList[p[-x, n - 1], x], {n, 0, 16}]]
(* Jean-François Alcover, Jun 20 2011 *)
Flatten[Map[CoefficientList[#,x]&, Table[Sum[Binomial[t - i, i] x^(i) (-1)^i, {i, 0, t}], {t, 1,15}]]] (* Rigoberto Florez, Aug 28 2022 *)

import math
L1 = [math.comb(t - i, i)*(-1)**i for t in range(16) for i in range(t)]
L1 = list(filter((0)._ne_, L1))
print(L1) # Rigoberto Florez, Sep 03 2022

a(n, m) = ((-1)^(m))*binomial(n-1-m, m), n>=1, m=0..ceiling(n/2)-1.
a(n, m) = [x^m]P(n,x), n>=1, m=0..ceiling(n/2)-1, with P(n,x) given above in terms of Chebyshev's S-polynomials.
P(n,x) = (u^(2*n) - v^(2*n))/(u^2 - v^2), where u and v are defined by u^2 + v^2 =1 and u*v = sqrt(x). Example: P(3,x) = (u^6 - v^6)/(u^2 - v^2) = u^4 + u^2*v^2 + v^4 = 1 - x. - Emeric Deutsch, Jun 16 2011
G.f.: 1/(1- x + y*x^2) = R(0)/2, where R(k) = 1 + 1/(1 - (2*k+1- x*y)*x/((2*k+2- x*y)*x + 1/R(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 09 2013
T(n, k) = GegenbauerC(k, (n+1)/2-k, -1) assuming the triangle (0,0) based. - Peter Luschny, May 10 2016

A119900 Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 4, 4, 0, 0, 1, 10, 5, 0, 0, 0, 6, 20, 6, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 8, 56, 56, 8, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 10, 120, 252, 120, 10, 0, 0, 0, 0, 0, 1, 55, 330, 462, 165, 11, 0, 0, 0, 0, 0, 0, 12, 220, 792, 792, 220, 12, 0, 0, 0, 0, 0, 0, 1, 78

Offset: 0

Emeric Deutsch, May 27 2006

Sum of terms in row n is 2^n (A000079). Sum of terms in column k is A001906(k+1) (the even-indexed Fibonacci numbers). Row n contains 1+floor(n/2) nonzero terms. Sum_{k=0..n} k*T(n,k) = (3n+1)*2^(n-2) = A066373(n+1) for n>=1.
Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0, 0,...] DELTA [2,-1/2,1/2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 02 2008
From R. Bagula's comment in A053122 (cf. Damianou link), the columns of this array give the coefficients (mod sign) of the characteristic polynomials for the Cartan matrix of the root system A_n. - Tom Copeland, Oct 11 2014
Odd rows contain the Pascal triangle numbers A091042. See A034867 and A034839 for some relations to tan(x). - Tom Copeland, Oct 15 2014

			The binary word 1/0/01/01/1/1/01 has 7 strictly increasing runs.
T(5,3)=6 because we have 0/01/01, 01/0/01, 01/01/0, 01/1/01, 01/01/1 and 1/01/01 (the runs are separated by /).
Triangle starts:
  1;
  0,2;
  0,1,3;
  0,0,4,4;
  0,0,1,10,5;
  0,0,0,6,20,6;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
A. Collins et al., Binary words, n-color compositions and bisection of the Fibonacci numbers, Fib. Quarterly, 51 (2013), 130-136.
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv:1110.6620 [math.RT], 2014.
R. Zielinski, Induction and Analogy in a Problem of Finite Sums arXiv:1608.04006 [math.GM], 2016.

Cf. A000079, A001906, A066373, A034867.
Cf. A098158. - Philippe Deléham, Dec 02 2008
Cf. A000292, A000389, A000580, A001263, A049310, A053122, A134264, A207543.

/* triangle */ [[Binomial(n+1, 2*k-n): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Oct 22 2017

T:=(n,k)->binomial(n+1,2*k-n): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Table[Binomial[n + 1, 2 k - n], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 21 2016 *)

for(n=0,10, for(k=0,n, print1(binomial(n+1, 2*k-n), ", "))) \\ G. C. Greubel, Oct 22 2017

T(n,k) = binomial(n+1,2k-n).
G.f.: 1/(1 - 2*t*z - t*(1-t)*z^2).
T(n,k) = A034867(n,n-k)
From Tom Copeland, Sep 30 2011: (Start)
With K(x,t) = 1/{d/dx{x/[t-1+1/(1-x)]}} = [t-1+1/(1-x)]^2/{t-[x/(1-x)]^2}, the g.f. of A119900 = K(x*t,t)-t+1.
From formulas in A134264: K(x,t)d/dx is a generator for A001263. A refinement of A119900 to partition polynomials is given by umbralizing
  K(x,t) roughly as K(h.x,h_0) and precisely as in A134264 as
  W(x)= 1/{d/dx[f(x)]}=1/{d/dx[x/h(x)]}. (End)
T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2). - Philippe Deléham, Oct 02 2011
From Tom Copeland, Dec 07 2015: (Start)
An alternate o.g.f. is (1/(x*t)) {-1 + 1 / [1 - (1/t)[x*t/(1-x*t)]^2]} = Sum_{n>0} x^(2(n-1)+1) t^(n-1) / (1-t*x)^(2n) = x + 2t x^2 + (t+3t^2) x^3 + ... .
The n-th diagonal has elements binomial(2n+1+k,k), starting with k=0 for the first non-vanishing element, with o.g.f. (1-x)^(-2(n+1)). The first few subdiagonals are shifted versions of A000292, A000389, and A000580. Cf. A049310.
See A034867 for the matrix representation for the infinitesimal generator K(x,t) d/dx for the Narayana polynomials. (End)
From Peter Bala, Aug 17 2016: (Start)
Let S(k,n) = Sum_{i = 1..n} i^k. Calculations in Zielinski 2016 suggest the following identity holds involving the p-th row elements of this triangle:
Sum_{k = 0..p} T(p,k)*S(2*k + 1,n) = (n*(n + 1)/2)^(p+1).
For example, for row 6 we find S(7,n) + 21*S(9,n) + 35*S(11,n) + 7*S(13,n) = (n*(n + 1)/2)^7.
There appears to be a similar result for the even power sums S(2*k,n) involving A207543. (End)

Keyword tabl added by Philippe Deléham, Jan 26 2010

A053123 Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order).

Original entry on oeis.org

1, 1, -2, 1, -4, 3, 1, -6, 10, -4, 1, -8, 21, -20, 5, 1, -10, 36, -56, 35, -6, 1, -12, 55, -120, 126, -56, 7, 1, -14, 78, -220, 330, -252, 84, -8, 1, -16, 105, -364, 715, -792, 462, -120, 9, 1, -18, 136, -560, 1365, -2002, 1716, -792, 165, -10, 1, -20, 171, -816, 2380, -4368, 5005, -3432, 1287, -220, 11, 1

Offset: 0

Wolfdieter Lang

T(n,m) = A053122(n,n-m).
G.f. for row polynomials and row sums same as in A053122.
Unsigned column sequences are A000012, A005843, A014105, A002492 for m=0..3, resp. and A053126-A053131 for m=4..9.
This is also the coefficient triangle for Chebyshev's U(2*n+1,x) polynomials expanded in decreasing odd powers of (2*x): U(2*n+1,x) = Sum_{m=0..n} T(n,m)*(2*x)^(2*(n-m)+1). See the W. Lang link given in A053125.
Unsigned version is mirror image of A078812. - Philippe Deléham, Dec 02 2008

			Triangle begins:
  1;
  1,  -2;
  1,  -4,  3;
  1,  -6, 10,   -4;
  1,  -8, 21,  -20,   5;
  1, -10, 36,  -56,  35,  -6;
  1, -12, 55, -120, 126, -56, 7; ...
E.g. fourth row (n=3) {1,-6,10,-4} corresponds to polynomial S(3,x-2) = x^3-6*x^2+10*x-4.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
Stephen Barnett, "Matrices: Methods and Applications", Oxford University Press, 1990, p. 132, 343.

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].
Index entries for sequences related to Chebyshev polynomials.

Cf. A053124, A053122, A172431.

Flat(List([0..10], n-> List([0..n], k-> (-1)^k*Binomial(2*n-k+1,k) ))); # G. C. Greubel, Jul 23 2019

[(-1)^k*Binomial(2*n-k+1,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 23 2019

A053123 := proc(n,m)
    (-1)^m*binomial(2*n+1-m,m) ;
end proc: # R. J. Mathar, Sep 08 2013

T[n_, m_]:= (-1)^m*Binomial[2*n+1-m, m]; Table[T[n, m], {n, 0, 11}, {m, 0, n}]//Flatten (* Jean-François Alcover, Mar 05 2014, after R. J. Mathar *)

for(n=0,10, for(k=0,n, print1((-1)^k*binomial(2*n-k+1,k), ", "))) \\ G. C. Greubel, Jul 23 2019

[[(-1)^k*binomial(2*n-k+1,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 23 2019

T(n, m) = 0 if n

T(n, m) = -2*T(n-1, m-1) + T(n-1, m) - T(n-2, m-2), T(n, -2) = 0, T(-2, m) = 0, T(n, -1) = 0 = T(-1, m), T(0, 0) = 1, T(n, m) = 0 if n

G.f. for m-th column (signed triangle): ((-1)^m)*x^m*Po(m+1, x)/(1-x)^(m+1), with Po(k, x) := Sum_{j=0..floor(k/2)} binomial(k, 2*j+1)*x^j.

The n-th degree polynomial is the characteristic equation for an n X n tridiagonal matrix with (diagonal = all 2's, sub and superdiagonals all -1's and the rest 0's), exemplified by the 4X4 matrix M = [2 -1 0 0 / -1 2 -1 0 / 0 -1 2 -1 / 0 0 -1 2]. - Gary W. Adamson, Jan 05 2005

Sum_{m=0..n} T(n,m)*(c(n))^(2*n-2*m) = 1/c(n), where c(n) = 2*cos(Pi/(2*n+3)). - L. Edson Jeffery, Sep 13 2013

A127677 Scaled coefficient table for Chebyshev polynomials 2T(2n, sqrt(x)/2) (increasing even scaled powers, without zero entries).

Original entry on oeis.org

2, -2, 1, 2, -4, 1, -2, 9, -6, 1, 2, -16, 20, -8, 1, -2, 25, -50, 35, -10, 1, 2, -36, 105, -112, 54, -12, 1, -2, 49, -196, 294, -210, 77, -14, 1, 2, -64, 336, -672, 660, -352, 104, -16, 1, -2, 81, -540, 1386, -1782, 1287, -546, 135, -18, 1, 2, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1

Offset: 0

Wolfdieter Lang, Mar 07 2007

2*T(2*n,x) = Sum_{m=0..n} a(n,m)*(2*x)^(2*m).
Closely related to A284982, which has opposite signs and rows begin with 0 of alternating signs instead of +/2. - Eric W. Weisstein, Apr 07 2017
Bisection triangle of A127672 (without zero entries, even part). The odd part is ((-1)^(n-m))*A111125(n,m).
If the leading 2 is replaced by a 1 we get the essentially identical sequence A110162. - N. J. A. Sloane, Jun 09 2007
Also row n gives coefficients of characteristic polynomial of the Cartan matrix for the root system B_n (or, equally, C_n). - Roger L. Bagula, May 23 2007
From Wolfdieter Lang, Oct 04 2013: (Start)
This triangle a(n,m) is used to express the length ratio side/R given by s(4*n+2) = 2*sin(Pi/(4*n+2)) = 2*cos(2*n*Pi/(4*n+2)) in a regular (4*n+2)-gon, inscribed in a circle with radius R, in terms of rho(4*n+2) = 2*cos(Pi/4*n+2), the length ratio of (the smallest diagonal)/side (for n=2 there is no such diagonal).
s(4*n+2) = Sum_{m=0..n}a(n,m)*rho(4*n+2)^(2*m). This formula is needed to show that the total sum of all length ratios in a (4*n+2)-gon is an integer in the algebraic number field Q(rho(4*n+2)). Note that rho(4*n+2) has degree delta(4*n+2) = A055034(4*n+2). Therefore one has to take s(4*n+2) modulo C(4*n+2, x=rho(4*n+2)), the minimal polynomial of rho(4*n+2) (see A187360). Thanks go to Seppo Mustonen for asking me to look into this problem. See ((-1)^(n-m))*A111125(n,m) for the (4*n)-gon situation. (End)

			The triangle a(n,m) starts:
n\m  0    1    2     3     4     5     6     7    8   9  10 ...
0:   2
1:  -2    1
2:   2   -4    1
3:  -2    9   -6     1
4:   2  -16   20    -8     1
5:  -2   25  -50    35   -10     1
6:   2  -36  105  -112    54   -12     1
7:  -2   49 -196   294  -210    77   -14     1
8:   2  -64  336  -672   660  -352   104   -16    1
9:  -2   81 -540  1386 -1782  1287  -546   135  -18   1
10:  2 -100  825 -2640  4290 -4004  2275  -800  170 -20  1
... Reformatted and extended by _Wolfdieter Lang_, Nov 21 2012.
n=3: [-2,9,-6,1] stands for -2*1 + 9*(2*x)^2 -6*(2*x)^4 +1*(2*x)^6 = 2*(1+18*x^2-48*x^4+32*x^6) = 2*T(6,x).
(4*n+2)-gon side/radius s(4*n+2) as polynomial in rho(4*n+2) = smallest diagonal/side: n=0: s(2) = 2 (rho(2)=0); n=1: s(6) = -2 + rho(6)^2 = -2 + 3 = 1, (C(6,x) = x^2 - 3); n=2: s(10) = 2 - 4*rho(10)^2 + 1*rho(10)^4 = 2 - 4*rho(10)^2 + (5*rho(10)^2 - 5) = -3 + rho(10)^2, (C(10,x) = x^4 - 5*x^2 + 5). - _Wolfdieter Lang_, Oct 04 2013

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 62
Sigurdur Helgasson,Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S. :ISBN 0-8218-2848-7, 1978,p. 463.

Wolfdieter Lang, First 10 rows and more.
P. Damianou , On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
P. Damianou and C. Evripidou, Characteristic and Coxeter polynomials for affine Lie algebras, arXiv preprint arXiv:1409.3956 [math.RT], 2014.
Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart., Nov. 2005, pp. 359-370. See Table 1.6 (an unsigned version).
Eric Weisstein's World of Mathematics, Cartan Matrix
Eric Weisstein's World of Mathematics, Dynkin Diagram

Cf. A284982 (opposite signs and rows begin with 0).
Row sums (signed): -A061347(n+3) for n>=0.
Row sums (unsigned): A005248(n) = L(2*n), where L=Lucas.
Cf. A005248, A053122.
Cf. A263916, A111125, A208513, A034807, A127672.

T[n_, m_, d_] := If[ n == m, 2, If[n == d && m == d - 1, -2, If[(n == m - 1 || n == m + 1), -1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] a = Join[M[1], Table[CoefficientList[CharacteristicPolynomial[M[d], x], x], {d, 1, 10} ]] (* Roger L. Bagula, May 23 2007 *)
CoefficientList[2 ChebyshevT[2 Range[0, 10], Sqrt[x]/2], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[Table[(-1)^n LucasL[2 n, Sqrt[-x]], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)

a(n,m) = {if(n>=2, -2*a(n-1,m)+a(n-1,m-1)-a(n-2,m), if(n==0, if(m!=0,0,2), if(m==0,-2, if(m==1,1,0))))};
for(n=0,10,for(m=0,n,print1(a(n,m),", "))) \\ Hugo Pfoertner, Jul 19 2020

a(n,m) = 0 if n < m; a(n,0) = 2*(-1)^n; a(n,m) = ((-1)^(n+m))*n*binomial(n+m-1, 2*m-1)/m.
a(n,m) = 0 if n < m, a(0,0) = 2, a(n,m) = (-1)^(n-m)*(2*n/(n+m))*binomial(n+m, n-m), n >= 1. From Waring's formula applied to Chebyshev's T-polynomials. See also A110162. - Wolfdieter Lang, Nov 21 2012
The o.g.f. Sum_{n>=0} p(n,x)*z^n, n>=0, for the row polynomials p(n,x) := Sum_{m=0..n} a(n,m)*x^m is (2 + z*(2-x))/((z+1)^2 - z*x). Here p(n,x) = R(2*n,sqrt(x)) := 2*T(2*n,sqrt(x)/2) with Chebyshev's T-polynomials. For the R-polynomials see A127672. - Wolfdieter Lang, Nov 28 2012
From Tom Copeland, Nov 07 2015: (Start)
A logarithmic generator is 2*(1-log(1+x))-log(1-t*x/(1+x)^2) = 2 - log(1+(2-t)*x+x^2) = 2 + (-2 + t)*x + (2 - 4*t + t^2) x^2/2 + (-2 + 9*t - 6*t^2 + t^3) x^3/3 + ..., so a number of relations to the Faber polynomials of A263916 hold with p(0,x) = 2:
1) p(n,x) = F(n,(2-x),1,0,0,..)
2) p(n,x) = (-1)^n 2 + F(n,-x,2x,-3x,...,(-1)^n n*x)
3) p(n,x) = (-1)^n [2 + F(n,x,2x,3x,...,n*x)].
The unsigned array contains the partial sums of A111125 modified by appending a column of zeros, except for an initial two, to A111125. Then the difference of consecutive rows of unsigned A127677, further modified by appending an initial rows of zeros, generates the modified A111125. Cf. A208513 and A034807.
For relations among the characteristic polynomials of Cartan matrices of the Coxeter root groups, Chebyshev polynomials, cyclotomic polynomials, and the polynomials of this entry, see Damianou (p. 12, 20, and 21) and Damianou and Evripidou (p. 7).
See A111125 for a relation to the squares of the odd row polynomials here with the constant removed.
p(n,x)^2 = 2 + p(2*n,x). See also A127672. (End)
a(n,m) = -2*a(n-1,m) + a(n-1,m-1) - a(n-2,m) for n >= 2 with initial conditions a(0,0) = 2, a(1,0) = -2, a(1,1) = 1, a(0,m) = 0 for m != 0, a(1,m) = 0 for m != 0,1. - William P. Orrick, Jun 09 2020
p(n,x) = (x-2)*p(n-1,x) - p(n-2,x) for n >= 2. - William P. Orrick, Jun 09 2020

Definition corrected by Eric W. Weisstein, Apr 06 2017

A110162 Riordan array ((1-x)/(1+x), x/(1+x)^2).

Original entry on oeis.org

1, -2, 1, 2, -4, 1, -2, 9, -6, 1, 2, -16, 20, -8, 1, -2, 25, -50, 35, -10, 1, 2, -36, 105, -112, 54, -12, 1, -2, 49, -196, 294, -210, 77, -14, 1, 2, -64, 336, -672, 660, -352, 104, -16, 1, -2, 81, -540, 1386, -1782, 1287, -546, 135, -18, 1, 2, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1

Offset: 0

Paul Barry, Jul 14 2005

Inverse of Riordan array A094527. Rows sums are A099837. Diagonal sums are A110164. Product of Riordan array A102587 and inverse binomial transform (1/(1+x), x/(1+x)).
Coefficients of polynomials related to Cartan matrices of types C_n and B_n: p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2), with p(x,0) = 1; p(x,1) = 2-x; p(x,2) = x^2-4*x-2. - Roger L. Bagula, Apr 12 2008
From Wolfdieter Lang, Nov 16 2012: (Start)
The alternating row sums are given in A219233.
For n >= 1 the row polynomials in the variable x^2 are R(2*n,x):=2*T(2*n,x/2) with Chebyshev's T-polynomials. See A127672 and also the triangle A127677.
(End)
From Peter Bala, Jun 29 2015: (Start)
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x/(1 + x)^2 and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = (1 - 2*x + sqrt(1 - 4*x))/2. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

			Triangle T(n,k) begins:
m\k  0    1    2     3     4     5     6    7    8   9 10 ...
0:   1
1:  -2    1
2:   2   -4    1
3:  -2    9   -6     1
4:   2  -16   20    -8     1
5:  -2   25  -50    35   -10     1
6:   2  -36  105  -112    54   -12     1
7:  -2   49 -196   294  -210    77   -14    1
8:   2  -64  336  -672   660  -352   104  -16    1
9:  -2   81 -540  1386 -1782  1287  -546  135  -18   1
10:  2 -100  825 -2640  4290 -4004  2275 -800  170 -20  1
... Reformatted and extended by _Wolfdieter Lang_, Nov 16 2012
Row polynomial n=2: P(2,x) = 2 - 4*x + x^2. R(4,x):= 2*T(4,x/2) = 2 - 4*x^2 + x^4. For P and R see a comment above. - _Wolfdieter Lang_, Nov 16 2012.

G. C. Greubel, Rows n=0..100 of triangle, flattened
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
T. M. Richardson, The Reciprocal Pascal Matrix, arXiv:1405.6315 [math.CO], 2014.

Cf. A128411. See A127677 for an almost identical triangle.

Cf. A136674, A053122.

/* As triangle */ [[(-1)^(n-k)*(Binomial(n+k,n-k) + Binomial(n+k-1,n-k-1)): k in [0..n]]: n in [0.. 12]]; // Vincenzo Librandi, Jun 30 2015

Table[If[n==0 && k==0, 1, (-1)^(n-k)*(Binomial[n+k, n-k] + Binomial[n+k-1, n-k-1])], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 16 2018 *)

{T(n,k) = (-1)^(n-k)*(binomial(n+k,n-k) + binomial(n+k-1,n-k-1))};
for(n=0, 12, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 16 2018

[[(-1)^(n-k)*(binomial(n+k,n-k) + binomial(n+k-1,n-k-1)) for k in range(n+1)] for n in range(12)] # G. C. Greubel, Dec 16 2018

T(n,k) = (-1)^(n-k)*(C(n+k,n-k) + C(n+k-1,n-k-1)), with T(0,0) = 1. - Paul Barry, Mar 22 2007
From Wolfdieter Lang, Nov 16 2012: (Start)
O.g.f. row polynomials P(n,x) := Sum(T(n,k)*x^k, k=0..n): (1-z^2)/(1+(x-2)*z+z^2) (from the Riordan property).
O.g.f. column No. k: ((1-x)/(1+x))*(x/(1+x)^2)^k, k >= 0.
T(0,0) = 1, T(n,k) = (-1)^(n-k)*(2*n/(n+k))*binomial(n+k,n-k), n>=1, and T(n,k) = 0 if n < k. (From the Chebyshev T-polynomial formula due to Waring's formula.)
(End)
T(n,k) = -2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 29 2013

A115141 Convolution of A115140 with itself.

Original entry on oeis.org

1, -2, -1, -2, -5, -14, -42, -132, -429, -1430, -4862, -16796, -58786, -208012, -742900, -2674440, -9694845, -35357670, -129644790, -477638700, -1767263190, -6564120420, -24466267020, -91482563640, -343059613650, -1289904147324, -4861946401452, -18367353072152

Offset: 0

Wolfdieter Lang, Jan 13 2006

This is the so-called A-sequence for the Riordan triangles A053122, A110162, A129818, A158454 and signed A158909. For the notion of Z- and A-sequences for Riordan arrays see the W. Lang link under A006232 with details and references. Wolfdieter Lang, Dec 20 2010. [Revised, Nov 13 2012, Nov 22 2012 and Oct 22 2019]

a(n)*(-1)^n is the A-sequence for the Riordan triangle A111125. - Wolfdieter Lang, Jun 26 2011

			G.f. = 1 - 2*x - x^2 - 2*x^3 - 5*x^4 - 14*x^5 - 42*x^6 - 132*x^7 - 429*x^8 + ...

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

Cf. A000108, A115140, A115142 - A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-2*x+Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019

a[n_] := -First[ ListConvolve[ cc = Array[ CatalanNumber, n-1, 0], cc]]; a[0] = 1; a[1] = -2; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 21 2011 *)
CoefficientList[Series[(1-2*x+Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

{a(n) = if( n<1, n==0, -(n==1) -binomial( 2*n-2, n-1) / n)} /* Michael Somos, Mar 28 2012 */

((1-2*x+sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^2 = (1-x) - x*c(x) with the o.g.f. c(x) = (1-sqrt(1-4*x) )/(2*x) of A000108 (Catalan numbers).
a(0)=1, a(1)=-2, a(n) = -C(n-1), n>=2, with C(n):=A000108(n) (Catalan). The start [1, -2] is row n=2 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
The convolution inverse is A000108(x)^2. - Michael Somos, Mar 28 2012
REVERT transform is A069271. - Michael Somos, Mar 28 2012
EULER transform of -A060165. - Michael Somos, Mar 28 2012
D-finite with recurrence: n*a(n) +2*(-2*n+3)*a(n-1)=0. - R. J. Mathar, Feb 21 2020

A123027 Triangle of coefficients of (1 - x)^nU(n,-(3x - 2)/(2*x - 2)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, -2, 3, 3, -10, 8, -4, 22, -38, 21, 5, -40, 111, -130, 55, -6, 65, -256, 474, -420, 144, 7, -98, 511, -1324, 1836, -1308, 377, -8, 140, -924, 3130, -6020, 6666, -3970, 987, 9, -192, 1554, -6588, 16435, -25088, 23109, -11822, 2584, -10, 255, -2472, 12720, -39430, 77645, -98160, 77378, -34690, 6765

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 24 2006

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A053122(n,j)*x^j*(1 - x)^(n - j).

			Triangle begins:
     1;
    -2,    3;
     3,  -10,    8;
    -4,   22,   38,    21;
     5,  -40,  111,  -130,    55;
    -6,   65, -256,   474,  -420,    144;
     7,  -98,  511, -1324,  1836,  -1308,   377;
    -8,  140, -924,  3130, -6020,   6666, -3970,    987;
     9, -192, 1554, -6588, 16435, -25088, 23109, -11822, 2584;
     ... reformatted and extended. _Franck Maminirina Ramaharo_, Oct 10 2018

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind
Wikipedia, Chebyshev polynomials

Cf. A008310, A049310, A053122, A111006.

Cf. A122753, A123018, A123019, A123021, A123199, A123202, A123217, A123221.

b0 = Table[CoefficientList[ChebyshevU[n, x/2 -1], x], {n, 0, 10}];
Table[CoefficientList[Sum[b0[[m+1]][[n+1]]*x^n*(1-x)^(m-n), {n,0,m}], x], {m,0,10}]//Flatten
(* Alternative Adamson Matrix method *)
t[n_, m_] = If[n==m, 2, If[n==m-1 || n==m+1, 1, 0]];
M[d_] := Table[t[n, m], {n, d}, {m, d}];
a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]];
Table[CoefficientList[Sum[a[[m+1]][[n+1]]*x^n*(1-x)^(m-n), {n, 0, m}], x], {m, 0, 10}]//Flatten

A053122(n, k) := if n < k then 0 else ((-1)^(n - k))*binomial(n + k + 1,  2*k + 1)$
P(x, n) := expand(sum(A053122(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
T(n, k) := ratcoef(P(x, n), x, k)$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, hipow(P(x, n), x)))$ /* Franck Maminirina Ramaharo, Oct 10 2018 */

def A053122(n, k): return 0 if (nA053122(n, j)*x^j*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (0..12)]) # G. C. Greubel, Jul 15 2021

From Franck Maminirina Ramaharo, Oct 10 2018: (Start)
Row n = coefficients in the expansion of (1/sqrt((5*x - 4)*x))*(((3*x - 2 + sqrt((5*x - 4)*x))/2)^(n + 1) - ((3*x - 2 - sqrt((5*x - 4)*x))/2)^(n + 1)).
G.f.: 1/(1 + (2 - 3*x)*t + (1 - x)^2*t^2).
E.g.f.: exp(t*(3*x - 2)/2)*(sqrt((5*x - 4)*x)*cosh(t*sqrt((5*x - 4)*x)/2) + (3*x - 2)*sinh(t*sqrt((5*x - 4)*x)/2))/sqrt((5*x - 4)*x).
T(n,1) = (-1)^(n+1)*A006503(n).
T(n,n) = A001906(n+1). (End)

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 10 2018

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A030528 Triangle read by rows: a(n,k) = binomial(k,n-k).

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A078812 Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).

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A034867 Triangle of odd-numbered terms in rows of Pascal's triangle.

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A115139 Array of coefficients of polynomials related to integer powers of the generating function of Catalan numbers A000108.

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A119900 Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.

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A127677 Scaled coefficient table for Chebyshev polynomials 2T(2n, sqrt(x)/2) (increasing even scaled powers, without zero entries).

A123027 Triangle of coefficients of (1 - x)^nU(n,-(3x - 2)/(2*x - 2)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.