A256893 -id:A256893 - cp's OEIS frontend

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

1, 1, 0, 2, 1, 0, 4, 3, 0, 0, 8, 8, 1, 0, 0, 16, 20, 5, 0, 0, 0, 32, 48, 18, 1, 0, 0, 0, 64, 112, 56, 7, 0, 0, 0, 0, 128, 256, 160, 32, 1, 0, 0, 0, 0, 256, 576, 432, 120, 9, 0, 0, 0, 0, 0, 512, 1280, 1120, 400, 50, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 03 2011

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.
Skewed version of triangle in A200139.
Triangle without zeros: A207537.
For the version with negative odd numbered columns, which is Riordan ((1-x)/(1-2*x), -x^2/(1-2*x)) see comments on A028297 and A039991. - Wolfdieter Lang, Aug 06 2014
This is an example of a stretched Riordan array in the terminology of Section 2 of Corsani et al. - Peter Bala, Jul 14 2015

			The triangle T(n,k) begins:
  n\k      0     1     2     3     4    5   6  7 8 9 10 11 ...
  0:       1
  1:       1     0
  2:       2     1     0
  3:       4     3     0     0
  4:       8     8     1     0     0
  5:      16    20     5     0     0    0
  6:      32    48    18     1     0    0   0
  7:      64   112    56     7     0    0   0  0
  8:     128   256   160    32     1    0   0  0 0
  9:     256   576   432   120     9    0   0  0 0 0
  10:    512  1280  1120   400    50    1   0  0 0 0  0
  11:   1024  2816  2816  1232   220   11   0  0 0 0  0  0
  ...  reformatted and extended. - _Wolfdieter Lang_, Aug 06 2014

C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Columns include A011782, A001792, A001793, A001794, A006974, A006975, A006976.
Diagonals sums are in A052980.
Cf. A028297, A081265, A124182, A131577, A039991 (zero-columns deleted, unsigned and zeros appended).
Cf. A098158, A200139, A207537.
Cf. A028297 (signed version, zeros deleted). Cf. A034839.

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

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 for k<0 or for n

Sum_{k=0..n} T(n,k)^2 = A002002(n) for n>0.

Sum_{k=0..n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n), A087455(n), A146559(n), A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 respectively.

G.f.: (1-x)/(1-2*x-y*x^2). - Philippe Deléham, Mar 03 2012

From Peter Bala, Jul 14 2015: (Start)

Factorizes as A034839 * A007318 = (1/(1 - x), x^2/(1 - x)^2) * (1/(1 - x), x/(1 - x)) as a product of Riordan arrays.

T(n,k) = Sum_{i = k..floor(n/2)} binomial(n,2*i) *binomial(i,k). (End)

Name changed, keyword:easy added, crossrefs A028297 and A039991 added, and g.f. corrected by Wolfdieter Lang, Aug 06 2014

A093375 Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 6, 1, 5, 16, 18, 8, 1, 6, 25, 40, 30, 10, 1, 7, 36, 75, 80, 45, 12, 1, 8, 49, 126, 175, 140, 63, 14, 1, 9, 64, 196, 336, 350, 224, 84, 16, 1, 10, 81, 288, 588, 756, 630, 336, 108, 18, 1, 11, 100, 405, 960, 1470, 1512, 1050, 480, 135, 20, 1, 12

Offset: 1

Ralf Stephan, Apr 28 2004

Number of n-long m-ary words avoiding the pattern 1-1'2'.
T(n,n+1) = Sum_{i=1..n} T(n,i).
Exponential Riordan array [(1+x)e^x, x] as a number triangle. - Paul Barry, Feb 17 2009
From Peter Bala, Jul 22 2014: (Start)
Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/
having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A059298. (End)

			Array T(m,n) (with rows m >= 1 and columns n >= 1) begins as follows:
   1   1   1   1   1   1 ...
   2   4   6   8  10  12 ...
   3   9  18  30  45  63 ...
   4  16  40  80 140 224 ...
   5  25  75 175 350 630 ...
   ...
Triangle S(n,k) = T(n-k+1, k+1) begins
.n\k.|....0....1....2....3....4....5....6
= = = = = = = = = = = = = = = = = = = = =
..0..|....1
..1..|....2....1
..2..|....3....4....1
..3..|....4....9....6....1
..4..|....5...16...18....8....1
..5..|....6...25...40...30...10....1
..6..|....7...36...75...80...45...12....1
...

Muniru A Asiru, Antidiagonals, n=1..100 flattened
Yasemin Alp and E. Gokcen Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 7.
Sergey Kitaev and Toufik Mansour, Partially ordered generalized patterns and k-ary words, arXiv:math/0210023 [math.CO], 2002.
Wikipedia, Sheffer sequence.

Rows include A045943. Columns include A002411, A027810.
Main diagonal is A037965. Subdiagonals include A002457.
Antidiagonal sums are A001792.
See A103283 for a signed version.
Cf. A103406, A059298, A073107 (unsigned inverse).

nmax:=14;; T:=List([1..nmax],n->List([1..nmax],k->k*Binomial(n+k-2,n-1)));;
b:=List([2..nmax],n->OrderedPartitions(n,2));;
a:=Flat(List([1..Length(b)],i->List([1..Length(b[i])],j->T[b[i][j][1]][b[i][j][2]]))); # Muniru A Asiru, Aug 07 2018

nmax = 10;
T = Transpose[CoefficientList[# + O[z]^(nmax+1), z]& /@ CoefficientList[(1 - x z)/(1 - z - x z)^2 + O[x]^(nmax+1), x]];
row[n_] := T[[n+1, 1 ;; n+1]];
Table[row[n], {n, 0, nmax}] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

# uses[riordan_array from A256893]
riordan_array((1+x)*exp(x), x, 8, exp=true) # Peter Luschny, Nov 02 2019

Triangle = P*M, the binomial transform of the infinite bidiagonal matrix M with (1,1,1,...) in the main diagonal and (1,2,3,...) in the subdiagonal, and zeros elsewhere. P = Pascal's triangle as an infinite lower triangular matrix. - Gary W. Adamson, Nov 05 2006
From Peter Bala, Sep 20 2012: (Start)
E.g.f. for triangle: (1 + z)*exp((1 + x)*z) = 1 + (2 + x)*z + (3 + 4*x + x^2)*z^2/2! + ....
O.g.f. for triangle: (1 - x*z)/(1 - z - x*z)^2 = 1 + (2 + x)*z + (3 + 4*x + x^2)*z^2 + ....
The n-th row polynomial R(n,x) of the triangle equals (1+x)^n + n*(1+x)^(n-1) for n >= 0 and satisfies d/dx(R(n,x)) = n*R(n-1,x), as well as R(n,x+y) = Sum_{k = 0..n} binomial(n,k)*R(k,x)*y^(n-k). The row polynomials are a Sheffer sequence of Appell type.
Matrix inverse of the triangle is a signed version of A073107. (End)
From Tom Copeland, Oct 20 2015: (Start)
With offset 0 and D = d/dx, the raising operator for the signed row polynomials P(n,x) is RP = x - d{log[e^D/(1-D)]}/dD = x - 1 - 1/(1-D) =  x - 2 - D - D^2 + ..., i.e., RP P(n,x) = P(n+1,x).
The e.g.f. for the signed array is (1-t) * e^(-t) * e^(x*t).
From the Appell formalism, the row polynomials PI(n,x) of A073107 are the umbral inverse of this entry's row polynomials; that is, P(n,PI(.,x)) = x^n = PI(n,P(.,x)) under umbral composition. (End)
From Petros Hadjicostas, Nov 01 2019: (Start)
As a triangle, we let S(n,k) = T(n-k+1, k+1) = (n-k+1)*binomial(n, k) for n >= 0 and 0 <= k <= n. See the example below.
As stated above by Peter Bala, Sum_{n,k >= 0} S(n,k)*z^n*x^k = (1 - x*z)/(1 - z -x*z)^2.
Also, Sum_{n, k >= 0} S(n,k)*z^n*x^k/n! = (1+z)*exp((1+x)*z).
As he also states, the n-th row polynomial is R(n,x) = Sum_{k = 0..n} S(n, k)*x^k = (1 + x)^n + n*(1 + x)^(n-1).
If we define the signed triangle S*(n,k) = (-1)^(n+k) * S(n,k) = (-1)^(n+k) * T(n-k+1, k+1), as Tom Copeland states, Sum_{n,k >= 0} S^*(n,k)*t^n*x^k/n! = (1-t)*exp((1-x)*(-t)) = (1-t) * e^(-t) * e^(x*t).
Apparently, S*(n,k) = A103283(n,k).
As he says above, the signed n-th row polynomial is P(n,x) =  (-1)^n*R(n,-x) = (x - 1)^n - n*(x - 1)^(n-1).
According to Gary W. Adamson, P(n,x) is "the monic characteristic polynomial of the n X n matrix with 2's on the diagonal and 1's elsewhere." (End)

A111593 Triangle of tanh numbers.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, -2, 0, 1, 0, 0, -8, 0, 1, 0, 16, 0, -20, 0, 1, 0, 0, 136, 0, -40, 0, 1, 0, -272, 0, 616, 0, -70, 0, 1, 0, 0, -3968, 0, 2016, 0, -112, 0, 1, 0, 7936, 0, -28160, 0, 5376, 0, -168, 0, 1, 0, 0, 176896, 0, -135680, 0, 12432, 0, -240, 0, 1, 0, -353792, 0, 1805056, 0, -508640, 0, 25872

Offset: 0

Wolfdieter Lang, Aug 23 2005

Sheffer triangle associated to Sheffer triangle A060081.
For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854.
In the umbral calculus (see the S. Roman reference) this triangle would be called associated for (1,arctanh(y)).
Without the n=0 row and m=0 column and unsigned, this is the Jabotinsky triangle A059419.
The inverse matrix of A with elements a(n,m), n,m>=0, is A111594.
The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A060081, satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.
The row polynomials p(n,x) (defined above) have e.g.f. exp(x*tanh(y)).
Exponential Riordan array [1, tanh(x)], inverse of [1, arctanh(x)] which is A111594. - Paul Barry, May 30 2010
Also the Bell transform of A155585(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			Binomial convolution of row polynomials: p(3,x)= -2*x+x^3; p(2,x)=x^2, p(1,x)= x, p(0,x)= 1, together with those from A060081:
s(3,x)= -5*x+x^3; s(2,x)= -1+x^2, s(1,x)= x, s(0,x)= 1;
therefore -5*(x+y)+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = -2*y+y^3 + 3*x*y^2 + 3*(-1+x^2)*y + (-5*x+x^3).
From _Paul Barry_, May 30 2010: (Start)
Triangle begins:
  1;
  0,     1;
  0,     0,     1;
  0,    -2,     0,     1;
  0,     0,    -8,     0,     1;
  0,    16,     0,   -20,     0,     1;
  0,     0,   136,     0,   -40,     0,     1;
  0,  -272,     0,   616,     0,   -70,     0,     1;
  0,     0, -3968,     0,  2016,     0,  -112,     0,     1;
Production matrix begins:
  0,   1;
  0,   0,   1;
  0,  -2,   0,   1;
  0,   0,  -6,   0,   1;
  0,   0,   0, -12,   0,   1;
  0,   0,   0,   0, -20,   0,   1;
  0,   0,   0,   0,   0, -30,   0,   1;
  0,   0,   0,   0,   0,   0, -42,   0,   1;
  0,   0,   0,   0,   0,   0,   0, -56,   0,   1; (End)

W. Lang, First 10 rows.

Row sums: A003723. Unsigned row sums: A006229.
Cf. A059419, A060081, A111594.
Cf. A002378.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> 2^(n+1)*euler(n+1, 1), 9); # Peter Luschny, Jan 26 2016

t[0, 0] = 1; t[n_, m_] := Sum[ Binomial[k+m-1, m-1]*(k+m)!*(-1)^(k)*2^(n-k-m)*StirlingS2[n, k+m], {k, 0, n-m}]/m!; Table[t[n, m], {n, 0, 11}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013, after Vladimir Kruchinin *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[2^(#+1)*EulerE[#+1, 1]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

T(n,m):=if n=0 and m=0 then 1 else sum(binomial(k+m-1,m-1)*(k+m)!*(-1)^(k)*2^(n-k-m)*stirling2(n,k+m),k,0,n-m)/m!; /* Vladimir Kruchinin, Jun 09 2011 */

# uses[riordan_array from A256893]
riordan_array(1, tanh(x), 9, exp=true) # Peter Luschny, Apr 19 2015

E.g.f. for column m>=0: ((tanh(x))^m)/m!.
a(n, m) = coefficient of x^n of ((tanh(x))^m)/m!, n>=m>=0, else 0.
a(n, m) = a(n-1, m-1) - (m+1)*m*a(n-1, m+1), a(n, -1):=0, a(0, 0)=1, a(n, m)=0 for nT(n,m) = (Sum_{k=0..n-m} binomial(k+m-1,m-1)*(k+m)!*(-1)^k*2^(n-k-m)*stirling2(n,k+m))/m!, T(0,0)=1. - Vladimir Kruchinin, Jun 09 2011
With e.g.f. exp(x*tanh(t)) = sum(n>= 0, P(n,x)*t^n/n!), the lowering operator is L = arctanh(d/dx) = d/dx + (1/3)(d/dx)^3 + (1/5)(d/dx)^5 + ..., and the raising operator is R = x [1 - (d/dx)^2], where L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x), since the sequence is a binomial Sheffer sequence. - Tom Copeland, Oct 01 2015
The raising operator R = x - x D^2 in matrix form acting on an o.g.f. (formal power series) is the transpose of the production matrix M below. The linear term x is the diagonal of ones after transposition. The other transposed diagonal (A002378) comes from -x D^2 x^n = -n * (n-1) x^(n-1). Then P(n,x) = (1,x,x^2,..) M^n (1,0,0,..)^T. - Tom Copeland, Aug 17 2016

A110165 Riordan array (1/sqrt(1-6x+5x^2),(1-3x-sqrt(1-6x+5x^2))/(2x)).

Original entry on oeis.org

1, 3, 1, 11, 6, 1, 45, 30, 9, 1, 195, 144, 58, 12, 1, 873, 685, 330, 95, 15, 1, 3989, 3258, 1770, 630, 141, 18, 1, 18483, 15533, 9198, 3801, 1071, 196, 21, 1, 86515, 74280, 46928, 21672, 7210, 1680, 260, 24, 1, 408105, 356283, 236736, 119154, 44982, 12510, 2484, 333, 27, 1

Offset: 0

Paul Barry, Jul 14 2005

Columns include A026375, A026376 and A026377. Inverse is A110168. Rows sums are A110166. Diagonal sums are A110167.
From Peter Bala, Jan 09 2022: (Start)
This Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = (1 - 3*x - sqrt(1 - 6*x + 5*x^2))/(2*x) 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 + 3*x + 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)

			Rows begin
    1;
    3,   1;
   11,   6,   1;
   45,  30,   9,   1;
  195, 144,  58,  12,   1;
  873, 685, 330,  95,  15,   1;
Production array begins:
  3, 1;
  2, 3, 1;
  0, 1, 3, 1;
  0, 0, 1, 3, 1;
  0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 0, 1, 3, 1;
  ... - _Philippe Deléham_, Feb 08 2014

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.

seq(seq( coeff((x^2 + 3*x + 1)^n, x, n-k), k = 0..n ), n = 0..10); # Peter Bala, Jan 09 2022

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/Sqrt[1-6#+5#^2]&, (1-3#-Sqrt[1-6#+5#^2])/(2#)&, 10] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Number triangle T(n, k) = Sum_{j = 0..n} C(n, j)C(2j, j+k).

T(n,0) = 3*T(n-1,0) + 2*T(n-1,1), T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + T(n-1,k+1) for k > 0, T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 24 2014

A122896 Riordan array (1, (1 - x - sqrt(1 - 2x - 3x^2)) / (2*x)), a Riordan array for directed animals. Triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 5, 3, 1, 0, 9, 12, 9, 4, 1, 0, 21, 30, 25, 14, 5, 1, 0, 51, 76, 69, 44, 20, 6, 1, 0, 127, 196, 189, 133, 70, 27, 7, 1, 0, 323, 512, 518, 392, 230, 104, 35, 8, 1, 0, 835, 1353, 1422, 1140, 726, 369, 147, 44, 9, 1

Offset: 0

Paul Barry, Sep 18 2006

Also the convolution triangle of the Motzkin numbers A001006. - Peter Luschny, Oct 08 2022

			Triangle begins:
[0] 1;
[1] 0,   1;
[2] 0,   1,   1;
[3] 0,   2,   2,   1;
[4] 0,   4,   5,   3,   1;
[5] 0,   9,  12,   9,   4,   1;
[6] 0,  21,  30,  25,  14,   5,   1;
[7] 0,  51,  76,  69,  44,  20,   6,  1;
[8] 0, 127, 196, 189, 133,  70,  27,  7, 1;
[9] 0, 323, 512, 518, 392, 230, 104, 35, 8, 1.

Row sums are A005773, number of directed animals of size n.
Product of A007318 and this sequence is A122897.
Cf. A001006, A007318, A064189.

T := proc(n,k) option remember;
if k=0 then return 0^n fi; if k>n then return 0 fi;
T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) end:
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, Aug 17 2016
# Uses function PMatrix from A357368.
PMatrix(10, n -> simplify(hypergeom([1 -n/2, -n/2+1/2], [2], 4))); # Peter Luschny, Oct 08 2022

T[n_, n_] = 1; T[, 0] = 0; T[n, k_] /; 0, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

# uses[riordan_array from A256893]
riordan_array(1, (1-x-sqrt(1-2*x-3*x^2))/(2*x), 11) # Peter Luschny, Aug 17 2016

Inverse of Riordan array (1, x / (1 + x + x^2)).
T(n+1, k+1) = A064189(n, k). - Philippe Deléham, Apr 21 2007
Riordan array (1, x*m(x)) where m(x) is the g.f. of Motzkin numbers (A001006). - Philippe Deléham, Nov 04 2009

A064580 Triangle associated with rooted trees with a degree constraint (A036765).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 9, 14, 13, 1, 5, 14, 28, 40, 36, 1, 6, 20, 48, 87, 118, 104, 1, 7, 27, 75, 161, 273, 357, 309, 1, 8, 35, 110, 270, 536, 866, 1100, 939, 1, 9, 44, 154, 423, 951, 1782, 2772, 3441, 2905, 1, 10, 54, 208, 630, 1572, 3310, 5928, 8946, 10900, 9118

Offset: 0

Henry Bottomley, Sep 21 2001

Main diagonal is A036765. - Paul D. Hanna, Nov 18 2016

			Triangle begins:
  1;
  1,  1;
  1,  2,  2;
  1,  3,  5,  5;
  1,  4,  9, 14, 13;
  1,  5, 14, 28, 40, 36;
  ...

Columns include A000012, A000027, A000096.
Main diagonal is A036765.
The sequence of triangles A010054 (triangle indicator), A007318 (Pascal), A026300 (Motzkin), A064580, ... converges to the triangle A009766 (Catalan).
Row sums give A159772.

a[n_, k_] /; 0 <= k <= n = a[n, k] = a[n - 1, k] + a[n - 1, k - 1] + a[n - 1, k - 2] + a[n - 1, k - 3]; a[0, 0] = 1; a[, ] = 0;
Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2018 *)

# uses[riordan_array from A256893]
M = riordan_array(1, x/(1+x+x^2+x^3), 12).inverse()
for m in M[1:]:
    print([r for r in reversed(list(m)) if r != 0]) # Peter Luschny, Aug 17 2016

a(n, k) = a(n-1, k) + a(n-1, k-1) + a(n-1, k-2) + a(n-1, k-3) with a(0, 0)=1 and a(n, k)=0 if n < k or k < 0.

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007

A104578 A Padovan convolution triangle.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 3, 0, 1, 2, 3, 3, 4, 0, 1, 2, 6, 6, 4, 5, 0, 1, 3, 7, 12, 10, 5, 6, 0, 1, 4, 12, 16, 20, 15, 6, 7, 0, 1, 5, 17, 30, 30, 30, 21, 7, 8, 0, 1, 7, 24, 45, 60, 50, 42, 28, 8, 9, 0, 1, 9, 36, 70, 95, 105, 77, 56, 36, 9, 10, 0, 1, 12, 50, 111, 160, 175, 168, 112, 72

Offset: 0

Paul Barry, Mar 16 2005

A Padovan convolution triangle. See A000931 for the Padovan sequence.
Row sums are tribonacci numbers A000073(n+2). Antidiagonal sums are A008346. The first columns are A000931(n+3), A228577.
From Wolfdieter Lang, Oct 30 2018: (Start)
The alternating row sums give A001057(n+1), for n >= 0.
The inverse of this Riordan triangle is given in A319203.
The row polynomials R(n, x) := Sum_{k=0..n} T(n, k)*x^k, with R(-1, x) = 0, appear in the Cayley-Hamilton formula for nonnegative powers of a 3 X 3 matrix with Det M = sigma(3;3) = x1*x2*x3 = +1, sigma(3; 2) := x1*x2 + x1*x*3 + x2*x^3  = -1 and Tr M = sigma(3; 1) = x1 + x2 = x, where x1, x2, and  x3, are the eigenvalues of M, and sigma the elementary symmetric functions, as M^n = R(n-2, x)*M^2  + (R(n-3, x) + R(n-4, x))*M + R(n-3, x)*1_3, for n >= 3, where M^0 = 1_3 is the 3 X 3 unit matrix.
For the Cayley-Hamilton formula for 3 X 3 matrices with Det M = +1, sigma(3,2) = +1 and Tr(M) = x see A321196.
(End)

			From _Wolfdieter Lang_, Oct 30 2018: (Start)
The triangle T begins:
    n\k   0  1  2  3  4  5  6  7  8  9 10 ...
    --------------------------------------
    0:    1
    1:    0  1
    2:    1  0  1
    3:    1  2  0  1
    4:    1  2  3  0  1
    5:    2  3  3  4  0  1
    6:    2  6  6  4  5  0  1
    7:    3  7 12 10  5  6  0  1
    8:    4 12 16 20 15  6  7  0  1
    9:    5 17 30 30 30 21  7  8  0  1
   10:    7 24 45 60 50 42 28  8  9  0  1
   ...
Cayley-Hamilton formula for the tribonacci Q-matrix TQ(x) =[[x,1,1], [1,0,0], [0,1,0]] with Det(TQ) = +1, sigma(3, 2) = -1, and Tr(TQ) = x. For n = 3: TQ(x)^3 = R(1, x)*TQ(x)^2  + (R(0 x) + R(-1, x))*TQ(x) + R(0, x)*1_3 = x*TQ(x)^2 + TQ(x) + 1_3. For x = 1 see also A058265 (powers of the tribonacci constant).
Recurrence: T(6, 2) = T(5, 1) + T(4, 2) + T(3, 2) = 3 + 3 + 0 = 6.
Z- and A- recurrence with A319202 = {1, 0, 1, 1, -1, -3, 0, ...}:
  T(5, 0) = 0*1 + 1*2 + 1*3 + (-1)*0 + (-3)*1 = 2; T(5,2) = 1*2 + 0*3 + 1*0 + 1*1 = 3.
Boas-Buck type recurrence with b = {0, 2, 3, ...}: T(5, 2) = ((1+2)/(5-2)) * (3*1 + 2*0 + 0*3) = 1*3 = 3.
(End)

Tomislav Došlic and Luka Podrug, Tilings of a Honeycomb Strip and Higher Order Fibonacci Numbers, arXiv:2203.11761 [math.CO], 2022.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A000073, A000931, A001057, A001608, A008346, A058265, A228577, A319202, A319203 (inverse), A321196.

T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + T[n-2, k] + T[n-3, k]; T[0, 0] = 1; T[, ] = 0; Table[T[n, k], {n, 0, 12}, {k, 0, n}] (* Jean-François Alcover, Jun 11 2019 *)

# uses[riordan_array from A256893]
riordan_array( 1/(1 - x^2 - x^3), x/(1 - x^2 - x^3), 8) # Peter Luschny, Nov 09 2018

Riordan array (1/(1 - x^2 - x^3), x/(1 - x^2 - x^3)).
T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-3,k), T(0,0)=1, T(n,k)=0 if k > n or if k < n. - Philippe Deléham, Jan 08 2014
From Wolfdieter Lang, Oct 30 2018: (Start)
The Riordan property  T = (G(x), x*G(x)) with G(x)= 1/(1-x^2-x^3) implies the following.
G.f. of row polynomials R(n, x) is G(x,z) = 1/(1- x*z - z^2 - z^3).
G.f. of column sequence k:  x^k/(1 - x^2 - x^3)^(k+1), k >= 0.
Boas-Buck recurrence (see the Aug 10 2017 remark in A046521, also for the reference):
T(n, k) = ((k+1)/(n-k))*Sum_{j=k..n-1} b(n-1-j)*T(j, k), for n >= 1, k = 0,1, ..., n-1, and input T(n,n) = 1, for n >= 0. Here b(n) = [x^n]*(d/dx)log(G(x)) = A001608(n+1), for n >= 0.
Recurrences from the A- and Z- sequences (see the W. Lang link under A006232 with references), which are A(n) = A319202(n) and Z(n) = A(n+1).
  T(0, 0) = 1, T(n, k) = 0 for n < k, and
  T(n, 0) = Sum_{j=0..n-1} Z(j)*T(n-1, j), for n >= 1, and
  T(n, k) = Sum_{j=0..n-k} A(j)*T(n-1, k-1+j), for n >= m >= 1.
(End)

A109267 Riordan array (1/(1 - xc(x) - x^2c(x)^2), x*c(x)) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 9, 6, 3, 1, 29, 19, 10, 4, 1, 97, 63, 34, 15, 5, 1, 333, 215, 118, 55, 21, 6, 1, 1165, 749, 416, 201, 83, 28, 7, 1, 4135, 2650, 1485, 736, 320, 119, 36, 8, 1, 14845, 9490, 5355, 2705, 1220, 484, 164, 45, 9, 1, 53791, 34318, 19473, 9983, 4628, 1923, 703, 219, 55, 10, 1

Offset: 0

Paul Barry, Jun 24 2005

Inverse of Riordan array (1-x-x^2, x(1-x)), A109264. Row sums are A109262(n+1). Diagonal sums are A109268. Columns include A081696, A109262, A109263.

			Rows begin
   1;
   1,  1;
   3,  2,  1;
   9,  6,  3,  1;
  29, 19, 10,  4,  1;
  97, 63, 34, 15,  5,  1;

Muniru A Asiru, Table of n, a(n) for n = 0..5150
Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.

Row sums A109262, sums along shallow diagonals A109268, A081696 (column 0), A109262 (column 1), A109263 (column 2).

Cf. A000108, A109264.

Flat(List([0..10],n->List([0..n],k->Sum([0..n-k],i->(Fibonacci(i+1)-2*Fibonacci(i))*Binomial(2*n-k-i,n))))); # Muniru A Asiru, Feb 19 2018

A109267 := (n,k) -> add(-combinat:-fibonacci(i-2)*binomial(2*n-k-i,n), i=0..n-k):
seq(seq(A109267(n, k), k = 0..n), n = 0..10); # Peter Bala, Feb 18 2018

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

The production matrix M (deleting the zeros) is:
  1, 1;
  2, 1, 1;
  2, 1, 1, 1;
  2, 1, 1, 1, 1;
... such that the n-th row of the triangle is the top row of M^n. - Gary W. Adamson, Feb 16 2012
From Peter Bala, Feb 18 2018: (Start)
T(n,k) = Sum_{i = 0..n-k} (Fibonacci(i+1) - 2*Fibonacci(i))* binomial(2*n-k-i,n), 0 <= k <= n.
The n-th row polynomial of the row reverse triangle equals the n-th degree Taylor polynomial of the function (1 - 2*x)/((1 - x)*(1 - x - x^2)) * 1/(1 - x)^n about 0. For example, for n = 4, (1 - 2*x)/((1 - x)*(1 - x - x^2)) * 1/(1 - x)^4 = 1 + 4*x + 10*x^2 + 19*x^3 + 29*x^4 + O(x^5), giving (29, 19, 10, 4, 1) as row 4. (End)

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A238731 Riordan array ((1-2*x)/(1-3*x+x^2), x/(1-3*x+x^2)).

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A097808 Riordan array ((1+2x)/(1+x)^2, 1/(1+x)) read by rows.

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A201701 Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).

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A093375 Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.

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A111593 Triangle of tanh numbers.

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A110165 Riordan array (1/sqrt(1-6x+5x^2),(1-3x-sqrt(1-6x+5x^2))/(2x)).

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A122896 Riordan array (1, (1 - x - sqrt(1 - 2*x - 3*x^2)) / (2*x)), a Riordan array for directed animals. Triangle read by rows.

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A238731 Riordan array ((1-2x)/(1-3x+x^2), x/(1-3*x+x^2)).

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

A122896 Riordan array (1, (1 - x - sqrt(1 - 2x - 3x^2)) / (2*x)), a Riordan array for directed animals. Triangle read by rows.

A109267 Riordan array (1/(1 - xc(x) - x^2c(x)^2), x*c(x)) where c(x) is the g.f. of A000108.