A053121 -id:A053121 - cp's OEIS frontend

1, 4, 1, 13, 5, 1, 38, 18, 6, 1, 104, 56, 24, 7, 1, 272, 160, 80, 31, 8, 1, 688, 432, 240, 111, 39, 9, 1, 1696, 1120, 672, 351, 150, 48, 10, 1, 4096, 2816, 1792, 1023, 501, 198, 58, 11, 1, 9728, 6912, 4608, 2815, 1524, 699, 256, 69, 12, 1, 22784, 16640, 11520

Offset: 0

Wolfdieter Lang, May 26 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is (((1-z)^2)/(1-2*z)^3)/(1-x*z/(1-z)).
This is the third member of the family of Riordan-type matrices obtained from A007318(n,m) (Pascal's triangle read as lower triangular matrix) by repeated application of the prs-procedure.
The column sequences appear as A049611(n+1), A001793, A001788, A055580, A055581, A055582, A055583 for m=0..6.

			[0] 1
[1] 4, 1
[2] 13, 5, 1
[3] 38, 18, 6, 1
[4] 104, 56, 24, 7, 1
[5] 272, 160, 80, 31, 8, 1
[6] 688, 432, 240, 111, 39, 9, 1
[7] 1696, 1120, 672, 351, 150, 48, 10, 1
Fourth row polynomial (n = 3): p(3, x) = 38 + 18*x + 6*x^2 + x^3.

Cf. A007318, A055248, A055249. Row sums: A049612(n+1)= A055584(n, 0).

T := (n, k) -> binomial(n, k)*hypergeom([3, k - n], [k + 1], -1):
for n from 0 to 7 do seq(simplify(T(n, k)), k = 0..n) od; # Peter Luschny, Sep 23 2024

a(n, m)=sum(A055249(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m)= sum(a(j, m), j=m..n-1)+ A055249(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: (((1-x)^2)/(1-2*x)^3)*(x/(1-x))^m, m >= 0.

T(n, k) = binomial(n, k)*hypergeom([3, k - n], [k + 1], -1). - Peter Luschny, Sep 23 2024

A060922 Convolution triangle for Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 3, 1, 4, 6, 1, 7, 17, 9, 1, 11, 38, 39, 12, 1, 18, 80, 120, 70, 15, 1, 29, 158, 315, 280, 110, 18, 1, 47, 303, 753, 905, 545, 159, 21, 1, 76, 566, 1687, 2568, 2120, 942, 217, 24, 1, 123, 1039, 3612, 6666, 7043, 4311

Offset: 0

Wolfdieter Lang, Apr 20 2001

In the language of Shapiro et al. (see A053121 for the reference) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. G.f. for row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) is (1+2*z)/(1-(1+x)*z-(1+2*x)*z^2).
Row sums give A060925. Column sequences (without leading zeros) are, for m=0..6: A000032(n+1)= A000204(n+1) (Lucas), A004799(n+1), A060929-33.
Bisection of this triangle gives triangles A060923 (even part) and A060924 (odd part).
For the m-th column sequence (without leading zeros) one has: a(n+m,m)= (pL1(m,n)*L(n+2)+pL2(m,n)*L(n+1))/(m!*5^m), m >= 0, with the Lucas numbers L(n)=A000032(n), n >= 0 and the row polynomials pL1(n,x) := sum(A061188(n,m)*x^n,m=0..n) and pL2(n,x) := sum(A061189(n,m)*x^m,m=0..n).
Riordan array ((1+2*x)/(1-x-x^2), x*(1+2*x)/(1-x-x^2)). - Philippe Deléham, Jan 21 2014
T is the convolution triangle of A000204 (see A357368). - Peter Luschny, Oct 19 2022

			p(2,x) = 4+6*x+x^2.
Triangle begins:
1 ;
3, 1;
4, 6, 1;
7, 17, 9, 1;
11, 38, 39, 12, 1;
18, 80, 120, 70, 15, 1;
29, 158, 315, 280, 110, 18, 1;
47, 303, 753, 905, 545, 159, 21, 1;

Cf. A000032.

# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, A000204); # Peter Luschny, Oct 19 2022

a(n, m)=((n-m+1)*a(n, m-1)+2*(2*n-m)*a(n-1, m-1)+4*(n-1)*a(n-2, m-1))/(5*m), n >= m >= 1, a(n, 0)= A000204(n+1)= A000032(n+1).
G.f. for m-th column: ((1+2*x)/(1-x-x^2))* ((x*(1+2*x))/(1-x-x^2))^m.
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 21 2014

Example improved by Philippe Deléham, Jan 21 2014

A054450 Triangle of partial row sums of unsigned triangle A049310(n,m), n >= m >= 0 (Chebyshev S-polynomials).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 4, 4, 1, 1, 8, 8, 5, 5, 1, 1, 13, 12, 12, 6, 6, 1, 1, 21, 21, 17, 17, 7, 7, 1, 1, 34, 33, 33, 23, 23, 8, 8, 1, 1, 55, 55, 50, 50, 30, 30, 9, 9, 1, 1, 89, 88, 88, 73, 73, 38, 38, 10, 10, 1, 1, 144, 144, 138, 138, 103, 103, 47, 47, 11, 11, 1, 1

Offset: 0

Wolfdieter Lang, Apr 27 2000 and May 08 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is Fib(z)/(1-x*z/(1-z^2)) where Fib(x)=1/(1-x-x^2) = g.f. for A000045(n+1) (Fibonacci numbers without 0).

This is the first member of the family of Riordan-type matrices obtained from the unsigned convolution matrix A049310 by repeated application of the partial row sums procedure.

			Triangle begins as:
   1;
   1,  1;
   2,  1,  1;
   3,  3,  1,  1;
   5,  4,  4,  1,  1;
   8,  8,  5,  5,  1,  1;
  13, 12, 12,  6,  6,  1,  1;
  21, 21, 17, 17,  7,  7,  1,  1;
  34, 33, 33, 23, 23,  8,  8,  1,  1;
  55, 55, 50, 50, 30, 30,  9,  9,  1, 1;
  89, 88, 88, 73, 73, 38, 38, 10, 10, 1, 1;
  ...
Fourth row polynomial (n=3): p(3,x) = 3 + 3*x + x^2 + x^3.

Cf. A000027, A000045, A029907 (row sums), A038793, A038794, A038795, A038796.

Cf. A049310, A052952, A053121, A054451, A054453, A099571, A099572, A152948.

A049310:= func< n,k | ((n+k) mod 2) eq 0 select (-1)^(Floor((n+k)/2)+k)*Binomial(Floor((n+k)/2), k) else 0 >;
A054450:= func< n,k | (&+[Abs(A049310(n,j)): j in [k..n]]) >;
[A054450(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 25 2022

A049310[n_, k_]:= A049310[n, k]= If[n<0, 0, If[k==n, 1, A049310[n-1, k-1] - A049310[n-2, k] ]];
A054450[n_, k_]:= A054450[n, k]= Sum[Abs[A049310[n,j]], {j,k,n}];
Table[A054450[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 25 2022 *)

@CachedFunction
def A049310(n, k):
    if (n<0): return 0
    elif (k==n): return 1
    else: return A049310(n-1, k-1) - A049310(n-2, k)
def A054450(n,k): return sum( abs(A049310(n,j)) for j in (k..n) )
flatten([[A054450(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 25 2022

T(n, m) = Sum_{k=m..n} |A049310(n, k)| (sequence of partial row sums in column m).

Column m recursion: T(n, m) = Sum_{j=m..n} T(j-1, m)*|A049310(n-j, 0)| + |A049310(n, m)|, n >= m >= 0, a(n, m) := 0 if n

T(n, 0) = A000045(n+1).

T(n, 1) = A052952(n-1).

T(n, 2) = A054451(n-2).

Sum_{k=0..n} T(n, k) = A029907(n) = A054453(n, 0).

G.f. for column m: Fib(x)*(x/(1-x^2))^m, m >= 0, with Fib(x) = g.f. A000045(n+1).

The corresponding square array has T(n, k) = Sum_{j=0..floor(k/2)} binomial(n+k-j, j). - Paul Barry, Oct 23 2004

From G. C. Greubel, Jul 25 2022: (Start)

T(n, 3) = A099571(n-3).

T(n, 4) = A099572(n-4).

T(n, n) = T(n, n-1) = A000012(n).

T(n, n-2) = A000027(n), n >= 2.

T(n, n-3) = A000027(n), n >= 3.

T(n, n-4) = A152948(n), n >= 4.

T(n, n-5) = A152948(n), n >= 5.

T(n, n-6) = A038793(n), n >= 6.

T(n, n-8) = A038794(n), n >= 8.

T(n, n-10) = A038795(n), n >= 10.

T(n, n-12) = A038796(n), n >= 12. (End)

A104219 Triangle read by rows: T(n,k) is number of Schroeder paths of length 2n and having k peaks at height 1, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 11, 7, 3, 1, 45, 28, 12, 4, 1, 197, 121, 52, 18, 5, 1, 903, 550, 237, 84, 25, 6, 1, 4279, 2591, 1119, 403, 125, 33, 7, 1, 20793, 12536, 5424, 1976, 630, 176, 42, 8, 1, 103049, 61921, 26832, 9860, 3206, 930, 238, 52, 9, 1, 518859, 310954, 134913, 49912

Offset: 0

Emeric Deutsch, Mar 14 2005

A Schroeder path is a lattice path starting from (0,0), ending at a point on the x-axis, consisting only of steps U = (1,1), D = (1,-1) and H = (2,0) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318).
This is an example of a Riordan (lower triangular) matrix. See the Shapiro et al. reference quoted under A053121. More precisely, this ordinary convolution triangle belongs to the Bell subgroup of the Riordan group. In the Shapiro et al. notation this is a Bell matrix (g(x), x*g(x)) with g(x) = (1+x-sqrt(1-6*x+x^2))/(4*x), the o.g.f. of A001003(n), n >= 0.
The g.f. for the row polynomials p(n,x) = Sum_{k=0..n} a(n,k)*x^k is g(y)/(1-x*y*g(y)) = (1-2*x*y+y-sqrt(1-6*y+y^2))/(2*y*(2-x-x*y+x^2*y)).
Triangular array in A011117 transposed. - Philippe Deléham, Mar 16 2005

			Triangle starts:
  [0]   1;
  [1]   1,   1;
  [2]   3,   2,   1;
  [3]  11,   7,   3,  1;
  [4]  45,  28,  12,  4,  1;
  [5] 197, 121,  52, 18,  5, 1;
  [6] 903, 550, 237, 84, 25, 6, 1;
T(3,1)=7 because we have HH(UD),H(UD)H,(UD)HH,UUDD(UD),(UD)UUDD,(UD)UHD, and
UHD(UD) (the peaks UD at height 1 are shown between parentheses).
From _Philippe Deléham_, Dec 04 2015: (Start)
Production matrix begins:
   1,  1;
   2,  1,  1;
   4,  2,  1, 1;
   8,  4,  2, 1, 1;
  16,  8,  4, 2, 1, 1;
  32, 16,  8, 4, 2, 1, 1;
  64, 32, 16, 8, 4, 2, 1, 1; (End)

Peter Luschny, Row n for n = 0..44.
Shishuo Fu, Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes, preprint 2013; Combinatorics, Probability and Computing, Volume 24, Issue 1 (Honouring the Memory of Philippe Flajolet - Part 3) January 2015, pp. 354-372.

Row sums are the large Schroeder numbers (A006318). Column 0 yields the little Schroeder numbers (A001003).

Cf. A104967 (matrix inverse), A091370.

G:=2/(1+z+sqrt(1-6*z+z^2)-2*z*t): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 11 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form
# Alternatively:
T_row := proc(n) local c,f,s;
c := N -> hypergeom([1-N, N+2], [2], -1);
f := n -> 1+add(simplify(c(i))*x^i,i=1..n):
s := j -> coeff(series(f(j)/(1-x*t*f(j)),x,j+1),x,j):
seq(coeff(s(n),t,j),j=0..n) end:
seq(T_row(n),n=0..10); # Peter Luschny, Oct 30 2015

T[n_, k_] := (-1)^(n - k) Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, 2];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Peter Luschny, Jan 08 2018 *)

{T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff(2/(1+X+sqrt(1-6*X+X^2)-2*X*Y),n,x),k,y)} \\ Paul D. Hanna, Mar 30 2005

def A104219_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)+sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [prec(n, k) for k in (1..n)]
for n in (1..9): print(A104219_row(n)) # Peter Luschny, Mar 16 2016

G.f.: 2/(1+z+sqrt(1-6*z+z^2)-2*z*t).
Another version of the triangle T(n, k), 0 <= k <= n, read by rows; given by [0, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] = 1; 0, 1; 0, 1, 1; 0, 3, 2, 1; 0, 11, 7, 3, 1; 0, 45, 28, 12, 4, 1; ... where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 16 2005
a(n, k) = (k+1)*hypergeom([1-n+k, n+2], [2], -1) if n > k; a(n, n)=1; a(n, k)=0 if n < k. - Wolfdieter Lang, Sep 12 2005
a(n, k) = ((k+1)/(n-k))*Sum_{p=1..n-k} binomial(n-k, p)*binomial(n+p, p-1) if n > k; a(n, n)=1; a(n, k)=0 if n < k. Proof with Lagrange's inversion theorem based on eq. y = 1+x*(1-2/y) where y=1/g(x), with g(x) the o.g.f. of A001003(n), n >= 0. Use G(k;y):=1/y^(k+1), k >= 0 to find the coefficients a(n, k) of x^n of G(k;1/g(x)). For this method see also the Larcombe and French paper on Catalan convolutions quoted under A033184. - Wolfdieter Lang, Sep 12 2005
G.f.: 1/(1-x*y-x/(1-x-x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction). - Paul Barry, Feb 01 2009
T((m+1)*n+r-1,m*n+r-1)*r/(m*n+r) = Sum_{k=1..n} (k/n)*T((m+1)*n-k-1,m*n-1)*(r+k,r), n >= m > 1, also T(n-1,m-1) = (m/n)*Sum_{k=1..n-m+1} k*A001003(k-1)*T(n-k-1,m-2), n >= m > 1. - Vladimir Kruchinin, Mar 17 2011
T(n, k) = (-1)^(n - k)*binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], 2). - Peter Luschny, Jan 08 2018

A054458 Convolution triangle based on A001333(n), n >= 1.

Original entry on oeis.org

1, 3, 1, 7, 6, 1, 17, 23, 9, 1, 41, 76, 48, 12, 1, 99, 233, 204, 82, 15, 1, 239, 682, 765, 428, 125, 18, 1, 577, 1935, 2649, 1907, 775, 177, 21, 1, 1393, 5368, 8680, 7656, 4010, 1272, 238, 24, 1, 3363, 14641, 27312, 28548, 18358, 7506, 1946, 308, 27, 1

Offset: 0

Wolfdieter Lang, Apr 26 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The G.f. for the row polynomials p(n,x) (increasing powers of x) is LPell(z)/(1-x*z*LPell(z)) with LPell(z) given in 'Formula'.
Column sequences are A001333(n+1), A054459(n), A054460(n) for m=0..2.
Mirror image of triangle in A209696. - Philippe Deléham, Mar 24 2012
Subtriangle of the triangle given by (0, 3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012
Riordan array ((1+x)/(1-2*x-x^2), (x+x^2)/(1-2*x-x^2)). - Philippe Deléham, Mar 25 2012

			Fourth row polynomial (n=3): p(3,x)= 17+23*x+9*x^2+x^3.
Triangle begins :
  1
  3, 1
  7, 6, 1
  17, 23, 9, 1
  41, 76, 48, 12, 1
  99, 233, 204, 82, 15, 1
  239, 682, 765, 428, 125, 18, 1. - _Philippe Deléham_, Mar 25 2012
(0, 3, -2/3, -1/3, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins :
  1
  0, 1
  0, 3, 1
  0, 7, 6, 1
  0, 17, 23, 9, 1
  0, 41, 76, 48, 12, 1
  0, 99, 233, 204, 82, 15, 1
  0, 239, 682, 765, 428, 125, 15, 1. - _Philippe Deléham_, Mar 25 2012

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A002203(n+1)/2. Row sums: A055099(n).
Cf. A001333, A054459, A054460.
Cf. A040000, A001333, A055099, A126473, A126501, A126528.

a(n, m) := ((n-m+1)*a(n, m-1) + (2n-m)*a(n-1, m-1) + (n-1)*a(n-2, m-1))/(4*m), n >= m >= 1; a(n, 0)= A001333(n+1); a(n, m) := 0 if n

G.f. for column m: LPell(x)*(x*LPell(x))^m, m >= 0, with LPell(x)= (1+x)/(1-2*x-x^2) = g.f. for A001333(n+1).

G.f.: (1+x)/(1-2*x-y*x-x^2-y*x^2). - Philippe Deléham, Mar 25 2012

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 3 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 25 2012

Sum_{k=0..n} T(n,k)*x^k = A040000(n), A001333(n+1), A055099(n), A126473(n), A126501(n), A126528(n) for x = -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Mar 25 2012

A055587 Triangle with columns built from row sums of the partial row sums triangles obtained from Pascal's triangle A007318. Essentially A049600 formatted differently.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 8, 4, 1, 1, 16, 20, 13, 5, 1, 1, 32, 48, 38, 19, 6, 1, 1, 64, 112, 104, 63, 26, 7, 1, 1, 128, 256, 272, 192, 96, 34, 8, 1, 1, 256, 576, 688, 552, 321, 138, 43, 9, 1, 1, 512, 1280, 1696, 1520, 1002, 501, 190, 53, 10, 1, 1, 1024, 2816, 4096

Offset: 0

Wolfdieter Lang, May 30 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is 1/((1-z)*(1-x*z*(1-z)/(1-2*z))).

Column m (without leading zeros) is obtained from convolution of A000012 (powers of 1) with m-fold convoluted A011782.

			{1}; {1, 1}; {1, 2, 1}; {1, 4, 3, 1}; {1, 8, 8, 4, 1}; ...
Fourth row polynomial (n=3): p(3,x)= 1+4*x+3*x^2+x^3

Cf. A049600, column sequences are A000012 (powers of 1), A000079 (powers of 2), A001792, A049611, A049612, A055589, A055852-5 for m=0..9, row sums: A055588.

t[n_, k_] := Hypergeometric2F1[k, k-n, 1, -1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 05 2014, after Paul D. Hanna *)

{T(n,k) = if( n<0 || k<0, 0, polcoeff( polcoeff( 1 / ((1 - z) * (1 - x*z * (1 - z) / (1 - 2*z) + z * O(z^n) + x * O(x^k))), k), n))}; /* Michael Somos, Sep 30 2003 */

{T(n,k)=if(k>n||n<0||k<0,0,if(k==0||k==n,1, sum(j=0,n-k,binomial(n-k,j)*binomial(k+j-1,k-1)););)} (Hanna)

a(n, m)= Am(n, 0) if n >= m >= 0 and a(n, m) := 0 if nA007318) with the lower triangular matrix A007318 (Pascal triangle) and prs^(m) is the partial row sums (prs) mapping for triangular matrices applied m times. See e.g. A055584 for m=4.
G.f. for column m: (1/(1-x))*(x*(1-x)/(1-2*x))^m, m >= 0.
T(n, k) = sum_{j=0..n-k} C(n-k, j)*C(k+j-1, k-1). - Paul D. Hanna, Jan 14 2004

A108044 Triangle read by rows: right half of Pascal's triangle (A007318) interspersed with 0's.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 6, 0, 4, 0, 1, 0, 10, 0, 5, 0, 1, 20, 0, 15, 0, 6, 0, 1, 0, 35, 0, 21, 0, 7, 0, 1, 70, 0, 56, 0, 28, 0, 8, 0, 1, 0, 126, 0, 84, 0, 36, 0, 9, 0, 1, 252, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1, 0, 462, 0, 330, 0, 165, 0, 55, 0, 11, 0, 1, 924, 0, 792, 0, 495, 0, 220, 0, 66

Offset: 0

N. J. A. Sloane, Jun 02 2005

Column k has e.g.f. Bessel_I(k,2x). - Paul Barry, Mar 10 2010
T(n,k) is the number of binary sequences of length n in which the number of ones minus the number of zeros is k. If 2 divides(n+k), such a sequence will have (n+k)/2 ones and (n-k)/2 zeros. Since there are C(n,(n+k)/2) ways to choose the sequence entries that get a one, T(n,k)=binomial(n,(n+k)/2) whenever (n+k) is even and T(n,k)= 0 otherwise. See the example below in the example section. - Dennis P. Walsh, Apr 11 2012
T(n,k) is the number of walks on the semi-infinite integer line with n steps that end at k. The walks start at 0, move at each step either one to the left or one to the right, and never enter the region of negative k. [Walks with impenetrable wall at -1/2. Dyck excursions of n steps that end at level k.] The variant without the restriction of negative positions is A053121. - R. J. Mathar, Nov 02 2023

			Triangle begins:
  1
  0 1
  2 0 1
  0 3 0 1
  6 0 4 0 1
  0 10 0 5 0 1
  20 0 15 0 6 0 1
From _Paul Barry_, Mar 10 2010: (Start)
Production matrix is
  0, 1,
  2, 0, 1,
  0, 1, 0, 1,
  0, 0, 1, 0, 1,
  0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 0, 1 (End)
T(5,1)=10 since there are 10 binary sequences of length 5 in which the number of ones exceed the number of zeros by exactly 1, namely, 00111, 01011, 01101, 01110, 10011, 10101, 10110, 11001, 11010, and 11100. Also, T(5,2)=0 since there are no binary sequences in which the number of ones exceed the number of zeros by exactly 2. - _Dennis P. Walsh_, Apr 11 2012

Harvey P. Dale, Rows n = 0..125 of triangle, flattened
Paul Barry and Aoife Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, section 8.
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 14.
Robert S. Maier, Sheffer Polynomials and the s-ordering of Exponential Boson Operators, arXiv:2508.13094 [quant-ph], 2025. See p. 27.
Paul Peart and Wen-Jin Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
Louis W. Shapiro, Seyoum Getu, Wen-Jin Woan, and Leon C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A108045 (matrix inverse),

Cf. A204293, A357136, A000984 (column 0), A001700 (column 1), A001791 (column 2), A002054 (column 3)

import Data.List (intersperse)
a108044 n k = a108044_tabl !! n !! k
a108044_row n = a108044_tabl !! n
a108044_tabl = zipWith drop [0..] $ map (intersperse 0) a007318_tabl
-- Reinhard Zumkeller, May 18 2013

T:=proc(n,k) if n+k mod 2 = 0 then binomial(n,(n+k)/2) else 0 fi end: for n from 0 to 13 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Jun 05 2005

b[n_,k_]:=If[EvenQ[n+k],Binomial[n,(n+k)/2],0]; Flatten[Table[b[n,k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, May 05 2013 *)

Each entry is the sum of those in the previous row that are to its left and to its right.
Riordan array (1/sqrt(1-4*x^2), (1-sqrt(1-4*x^2))/(2*x)).
T(n, k) = binomial(n, (n+k)/2) if n+k is even, T(n, k)=0 if n+k is odd. G.f.=f/(1-tg), where f=1/sqrt(1-4x^2) and g=(1-sqrt(1-4x^2))/(2x). - Emeric Deutsch, Jun 05 2005
From Peter Bala, Jun 29 2015: (Start)
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = ( 1 - sqrt(1 - 4*x) )/(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 + 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) ).
The inverse array is A108045 (a hitting time array with h(x) = x/(1 + x^2)). (End)

More terms from Emeric Deutsch and Christian G. Bower, Jun 05 2005

A115193 Generalized Catalan triangle of Riordan type, called C(1,2).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 13, 13, 5, 1, 67, 67, 27, 7, 1, 381, 381, 157, 45, 9, 1, 2307, 2307, 963, 291, 67, 11, 1, 14589, 14589, 6141, 1917, 477, 93, 13, 1, 95235, 95235, 40323, 12867, 3363, 723, 123, 15, 1, 636925

Offset: 0

Wolfdieter Lang, Feb 23 2006

This triangle is the first of a family of generalizations of the Catalan convolution triangle A033184 (which belongs to the Bell subgroup of the Riordan group).
The o.g.f. of the row polynomials P(n,x):=Sum_{m=0..n} a(n,m)*x^n is D(x,z) = g(z)/(1 - x*z*c(2*z)) = g(z)*(2*z-x*z*(1-2*z*c(2*z)))/(2*z-x*z+(x*z)^2), with g(z) and c(z) defined below.
This is the Riordan triangle named (g(x),x*c(2*x)) with g(x):=(1+2*x*c(2*x))/(1+x) and c(x) is the o.g.f. of A000108 (Catalan numbers). g(x) is the o.g.f. of A064062 (C(2;n) Catalan generalization).
The column sequences (without leading zeros) are A064062, A064062(n+1), A084076, A115194, A115202-A115204, for m=0..6.
For general Riordan convolution triangles (lower triangular matrices) see the Shapiro et al. reference given in A053121.

			Triangle begins:
   1;
   1,  1;
   3,  3,  1;
  13, 13,  5,  1;
  67, 67, 27,  7,  1;
  ...
Production matrix begins:
    1,   1;
    2,   2,   1;
    4,   4,   2,   1;
    8,   8,   4,   2,   1;
   16,  16,   8,   4,   2,   1;
   32,  32,  16,   8,   4,   2,   1;
   64,  64,  32,  16,   8,   4,   2,   1;
  128, 128,  64,  32,  16,   8,   4,   2,   1;
  ... _Philippe Deléham_, Sep 22 2014

Nathaniel Johnston, Table of n, a(n) for n = 0..5150 (up to row 100)
Wolfdieter Lang, First 10 rows.

Row sums give A115197. Compare with the row reversed and scaled triangle A115195.

Cf. A116866 (similar sequence C(1,3)).

lim:=7: c:=(1-sqrt(1-8*x))/(4*x): g:=(1+2*x*c)/(1+x): gf1:=g*(x*c)^m: for m from 0 to lim do t:=taylor(gf1,x,lim+1): for n from 0 to lim do a[n,m]:=coeff(t,x,n):od:od: seq(seq(a[n,m],m=0..n),n=0..lim); # Nathaniel Johnston, Apr 30 2011

A110510[n_, k_] := (k/n)*Binomial[2*n - k - 1, n - k]*2^(n - k);
T[n_, k_] := If[n == 0, 1, Sum[A110510[n, i], {i, k, n}]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 17 2025 *)

G.f. for column m>=0 is g(x)*(x*c(2*x))^m, with g(x):=(1+2*x*c(2*x))/(1+x) and c(x) is the o.g.f. of A000108 (Catalan numbers).

T(n,k) = Sum_{i=k..n} A110510(n,i) for 0 <= k <= n. - Werner Schulte, Mar 24 2019

A189231 Extended Catalan triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 2, 8, 3, 4, 1, 10, 5, 15, 4, 5, 1, 5, 30, 9, 24, 5, 6, 1, 35, 14, 63, 14, 35, 6, 7, 1, 14, 112, 28, 112, 20, 48, 7, 8, 1, 126, 42, 252, 48, 180, 27, 63, 8, 9, 1, 42, 420, 90, 480, 75, 270, 35, 80, 9, 10, 1, 462, 132, 990, 165, 825, 110, 385, 44, 99, 10, 11, 1

Offset: 0

Peter Luschny, May 01 2011

Let S(n,k) denote the coefficients of the positive powers of the Laurent polynomials C_n(x) = (x+1/x)^(n-1)*(x-1/x)*(x+1/x+n) (if n>0) and C_0(x) = 0.
Then T(n,k) = S(n+1,k+1) for n>=0, k>=0.
The classical Catalan triangle A053121(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is odd.
The complementary Catalan triangle A189230(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is even.
T(n,0) are the extended Catalan numbers A057977(n).

			The Laurent polynomials:
C(0,x) =                 0
C(1,x) =               x - 1/x
C(2,x) =         x^2 + x - 1/x - 1/x^2
C(3,x) = x^3 + 2 x^2 + x - 1/x - 2/x^2 -1/x^3
Triangle T(n,k) = S(n+1,k+1) starts
[0]   1,
[1]   1,  1,
[2]   1,  2,  1,
[3]   3,  2,  3,  1,
[4]   2,  8,  3,  4,  1,
[5]  10,  5, 15,  4,  5,  1,
[6]   5, 30,  9, 24,  5,  6,  1,
[7]  35, 14, 63, 14, 35,  6,  7, 1,
    [0],[1],[2],[3],[4],[5],[6],[7]

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, The lost Catalan numbers

Cf. A053121, A162246, A057977, A189230.

A189231_poly := (n,x)-> `if`(n=0,0,(x+1/x)^(n-2)*(x-1/x)*(x+1/x+n-1)):
seq(print([n],seq(coeff(expand(A189231_poly(n,x)),x,k),k=1..n)),n=1..9);
A189231 := proc(n,k) option remember; `if`(k>n or k<0, 0, `if`(n=k, 1, A189231(n-1,k-1)+modp(n-k,2)*A189231(n-1,k)+A189231(n-1,k+1))) end:
seq(print(seq(A189231(n,k),k=0..n)),n=0..9);

t[n_, k_] /; (k > n || k < 0) = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + Mod[n-k, 2]*t[n-1, k] + t[n-1, k+1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2013 *)

Recurrence: If k>n or k<0 then T(n,k) = 0 else if n=k then T(n,k) = 1; otherwise T(n,k) = T(n-1,k-1) + ((n-k) mod 2)*T(n-1,k) + T(n-1,k+1).
S(n,k) = (k/n)* A162246(n,k) for n>0 where S(n,k) are the coefficients from the definition provided the triangle A162246 is indexed in Laurent style by the recurrence: if abs(k) > n then A162246(n,k) = 0 else if n = k then A162246(n,k) = 1 and otherwise A162246(n,k) = A162246(n-1,k-1)+ modp(n-k,2) * A162246(n-1,k) + A162246(n-1,k+1).
Row sums: A189911(n) = A162246(n,n) + A162246(n,n+1) for n>0.

A363718 Irregular triangle read by rows. An infinite binary tree which has root node 1 in row n = 0. Each node then has left child m-1 if greater than 0 and right child m+1, where m is the value of the parent node.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 1, 3, 1, 3, 3, 5, 2, 2, 4, 2, 2, 4, 2, 4, 4, 6, 1, 3, 1, 3, 3, 5, 1, 3, 1, 3, 3, 5, 1, 3, 3, 5, 3, 5, 5, 7, 2, 2, 4, 2, 2, 4, 2, 4, 4, 6, 2, 2, 4, 2, 2, 4, 2, 4, 4, 6, 2, 2, 4, 2, 4, 4, 6, 2, 4, 4, 6, 4, 6, 6, 8, 1, 3, 1, 3, 3, 5, 1, 3, 1

Offset: 0

John Tyler Rascoe, Jun 17 2023

The paths through the tree represent the compositions counted in A173258 that have first part 1.
For rows n > 1, row n starts with row n-2.
Any positive number k will first appear in the (k-1)-th row and thereafter in rows of opposite parity to k. The number of times k will appear in row n is A053121(n,k-1).
Row n >= 1 gives the row lengths of the Christmas tree pattern of order n (cf. A367508). - Paolo Xausa, Nov 26 2023
A new row can be generated by applying the morphism 1 -> 2, t -> {t-1,t+1} (for t > 1) to the previous row. - Paolo Xausa, Dec 08 2023

			Triangle begins:
  n=0:  1;
  n=1:  2;
  n=2:  1, 3;
  n=3:  2, 2, 4;
  n=4:  1, 3, 1, 3, 3, 5;
  n=5:  2, 2, 4, 2, 2, 4, 2, 4, 4, 6;
  n=6:  1, 3, 1, 3, 3, 5, 1, 3, 1, 3, 3, 5, 1, 3, 3, 5, 3, 5, 5, 7;
  ...
The binary tree starts with root 1 in row n = 0. In row n = 2, the parent node 2 has the first left child since 2 - 1 > 0.
The tree begins:
row
[n]
[0]                   1
                       \
[1]            _________2_________
              /                   \
[2]          1                _____3_____
              \              /           \
[3]          __2__        __2__         __4__
            /     \      /     \       /     \
[4]        1       3    1       3     3       5
            \     / \    \     / \   / \     / \
[5]          2   2   4    2   2   4 2   4   4   6
.

Paolo Xausa, Table of n, a(n) for n = 0..13494 (rows 0..15 of the triangle, flattened).

Cf. A001405 (row lengths), A000079 (row sums).

Cf. A000108, A007001, A053121, A173258, A367508, A367951, A367953.

SubstitutionSystem[{1->{2},t_/;t>1:>{t-1,t+1}},{1},8] (* Paolo Xausa, Dec 23 2023 *)

def A363718_rowlist(root,row):
    A = [[root]]
    for i in range(0,row):
        A.append([])
        for j in range(0,len(A[i])):
            if A[i][j] != 1:
                A[i+1].append(A[i][j]-1)
            A[i+1].append(A[i][j]+1)
    return(A)
A363718_rowlist(1, 8)

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A055252 Triangle of partial row sums (prs) of triangle A055249.

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A060922 Convolution triangle for Lucas numbers A000032(n+1), n >= 0.

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A054450 Triangle of partial row sums of unsigned triangle A049310(n,m), n >= m >= 0 (Chebyshev S-polynomials).

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A104219 Triangle read by rows: T(n,k) is number of Schroeder paths of length 2n and having k peaks at height 1, for 0 <= k <= n.

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A054458 Convolution triangle based on A001333(n), n >= 1.

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A055587 Triangle with columns built from row sums of the partial row sums triangles obtained from Pascal's triangle A007318. Essentially A049600 formatted differently.

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A108044 Triangle read by rows: right half of Pascal's triangle (A007318) interspersed with 0's.

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Haskell