A321620 -id:A321620 - cp's OEIS frontend

A321624 The Riordan square of the Lucas numbers, triangle read by rows, T(n, k) for 0 <= k <= n.

1, 1, 1, 3, 4, 1, 4, 10, 7, 1, 7, 24, 26, 10, 1, 11, 49, 77, 51, 13, 1, 18, 98, 200, 190, 85, 16, 1, 29, 187, 473, 595, 390, 128, 19, 1, 47, 350, 1056, 1658, 1450, 704, 180, 22, 1, 76, 642, 2253, 4255, 4688, 3062, 1159, 241, 25, 1

Offset: 0

Peter Luschny, Nov 22 2018

Compare A000032 (Lucas numbers with a(0) = 2), A000204 (Lucas numbers with a(0) undefined). Our variant has a(0) = 1.

Triangle, read by rows, given by [1, 2, -5/2, 1/2, 0, 0, 0, 0, 0, ...]DELTA[1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 06 2020

			[0] [  1]
[1] [  1,    1]
[2] [  3,    4,    1]
[3] [  4,   10,    7,    1]
[4] [  7,   24,   26,   10,    1]
[5] [ 11,   49,   77,   51,   13,    1]
[6] [ 18,   98,  200,  190,   85,   16,    1]
[7] [ 29,  187,  473,  595,  390,  128,   19,   1]
[8] [ 47,  350, 1056, 1658, 1450,  704,  180,  22,   1]
[9] [ 76,  642, 2253, 4255, 4688, 3062, 1159, 241,  25, 1]

T(n, 0) = A000204, A000032 (Lucas), A321573 (row sums), A000007 (alternating row sums).

Cf. A321620.

# The function RiordanSquare is defined in A321620.
Lucas :=  1 + x*(1 + 2*x)/(1 - x - x^2); RiordanSquare(Lucas, 10);

(* The function RiordanSquare is defined in A321620. *)
Lucas = 1 + x*(1 + 2*x)/(1 - x - x^2);
RiordanSquare[Lucas, 10] (* Jean-François Alcover, Jun 15 2019, from Maple *)

# uses[riordan_square from A321620]
riordan_square(1 + x*(1 + 2*x)/(1 - x - x^2), 10)

T(0,0) = 1, T(1,1) = 1, T(1,0) = 1, T(n,k) = 0 for k<0 and for k>n, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k) + 2*T(n-2,k-1), for n>1. - Philippe Deléham, Feb 06 2020

A321625 The Riordan square of the swinging factorial (A056040), triangle read by rows, T(n, k) for 0 <= k<= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 10, 5, 1, 6, 22, 22, 7, 1, 30, 66, 66, 38, 9, 1, 20, 140, 218, 146, 58, 11, 1, 140, 372, 574, 542, 270, 82, 13, 1, 70, 826, 1680, 1708, 1134, 446, 110, 15, 1, 630, 1930, 4156, 5432, 4126, 2106, 682, 142, 17, 1

Offset: 0

Peter Luschny, Nov 22 2018

			[0] [   1]
[1] [   1,    1]
[2] [   2,    3,    1]
[3] [   6,   10,    5,    1]
[4] [   6,   22,   22,    7,    1]
[5] [  30,   66,   66,   38,    9,    1]
[6] [  20,  140,  218,  146,   58,   11,    1]
[7] [ 140,  372,  574,  542,  270,   82,   13,   1]
[8] [  70,  826, 1680, 1708, 1134,  446,  110,  15,  1]
[9] [ 630, 1930, 4156, 5432, 4126, 2106,  682, 142, 17,  1]

T(n, 0) = A056040 (swinging factorial), A321626 (row sums), A000007 (alternating row sums).

Cf. A321620.

# The function RiordanSquare is defined in A321620.
SwingingFactorial := (1 + x/(1 - 4*x^2))/sqrt(1 - 4*x^2);
RiordanSquare(SwingingFactorial, 10);

(* The function RiordanSquare is defined in A321620. *)
SwingingFactorial = (1 + x/(1 - 4*x^2))/Sqrt[1 - 4*x^2];
RiordanSquare[SwingingFactorial, 10] (* Jean-François Alcover, Jun 15 2019, from Maple *)

# uses[riordan_square from A321620]
riordan_square((1 + x/(1 - 4*x^2))/sqrt(1 - 4*x^2), 10)

A321627 The Riordan square of the double factorial of odd numbers. Triangle T(n, k), 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 15, 21, 7, 1, 105, 144, 48, 10, 1, 945, 1245, 372, 84, 13, 1, 10395, 13140, 3357, 726, 129, 16, 1, 135135, 164745, 35415, 6873, 1233, 183, 19, 1, 2027025, 2399040, 434520, 73116, 12306, 1920, 246, 22, 1

Offset: 0

Peter Luschny, Dec 07 2018

The Riordan square is defined in A321620.

Triangle, read by rows, given by [1, 2, 3, 4, 5, 6, 7, …] DELTA [1, 0, 0, 0, 0, 0, 0, 0, …] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 17 2020

			Triangle starts:
[0][     1]
[1][     1,      1]
[2][     3,      4,     1]
[3][    15,     21,     7,    1]
[4][   105,    144,    48,   10,    1]
[5][   945,   1245,   372,   84,   13,   1]
[6][ 10395,  13140,  3357,  726,  129,  16,  1]
[7][135135, 164745, 35415, 6873, 1233, 183, 19, 1]

First column are the double factorial of odd numbers A001147.
Second column is number of singletons in pair-partitions A233481.
Row sums are A321628, alternating row sums are A000007.
Cf. A321620.

# The function RiordanSquare is defined in A321620.
cf := proc(dim) local k, m; m := 1;
for k from dim by -1 to 1 do m := 1 - k*x/m od;
1/m end: RiordanSquare(cf(9), 9);

(* The function RiordanSquare is defined in A321620. *)
cf[dim_] := Module[{k, m=1}, For[k=dim, k >= 1, k--, m = 1 - k*x/m]; 1/m];
RiordanSquare[cf[9], 9] (* Jean-François Alcover, Jun 15 2019, from Maple *)

A322944 Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 6, 1, 6, 38, 15, 1, 24, 272, 188, 28, 1, 120, 2200, 2340, 580, 45, 1, 720, 19920, 30280, 11040, 1390, 66, 1, 5040, 199920, 413560, 206920, 37450, 2842, 91, 1, 40320, 2204160, 5989760, 3931200, 955920, 102816, 5208, 120, 1

Offset: 0

Peter Luschny, Jan 02 2019

The polynomials represent a family of orthogonal polynomials which obey a recurrence of the form p(n, x) = (x+r(n))*p(n-1, x) - s(n)*p(n-2, x) + t(n)*p(n-3, x) - u(n)*p(n-4, x). For the details see the Maple program.

We conjecture that the polynomials have only negative and simple real roots.

			Triangle starts:
[0]    1;
[1]    1,      1;
[2]    2,      6,      1;
[3]    6,     38,     15,      1;
[4]   24,    272,    188,     28,     1;
[5]  120,   2200,   2340,    580,    45,    1;
[6]  720,  19920,  30280,  11040,  1390,   66,  1;
[7] 5040, 199920, 413560, 206920, 37450, 2842, 91, 1;
Production matrix starts:
   1;
   1,    1;
   3,    5,    1;
   6,   18,    9,    1;
   6,   42,   45,   13,    1;
   0,   48,  132,   84,   17,    1;
   0,    0,  180,  300,  135,   21,    1;
   0,    0,    0,  480,  570,  198,   25,    1;

Peter Luschny, Plot of the polynomials.

p(n, 1) = A322943(n) (row sums); p(n, 0) = n! = A000142(n).
A321966 (m=2), this sequence (m=3).
Cf. A321620.

P := proc(n) option remember; local a, b, c, d;
a := n -> 4*n-3; b := n -> 3*(n-1)*(2*n-3);
c := n -> (n-1)*(n-2)*(4*n-9); d := n -> (n-2)*(n-1)*(n-3)^2;
if n = 0 then return 1 fi;
if n = 1 then return x + 1 fi;
if n = 2 then return x^2 + 6*x + 2 fi;
if n = 3 then return x^3 + 15*x^2 + 38*x + 6 fi;
expand((x+a(n))*P(n-1) - b(n)*P(n-2) + c(n)*P(n-3) - d(n)*P(n-4)) end:
seq(print(P(n)), n=0..9); # Computes the polynomials.

a[n_] := 4n - 3;
b[n_] := 3(n - 1)(2n - 3);
c[n_] := (n - 1)(n - 2)(4n - 9);
d[n_] := (n - 2)(n - 1)(n - 3)^2;
P[n_] := P[n] = Switch[n, 0, 1, 1, x + 1, 2, x^2 + 6x + 2, 3, x^3 + 15x^2 + 38x + 6, _, Expand[(x + a[n]) P[n - 1] - b[n] P[n - 2] + c[n] P[n - 3] - d[n] P[n - 4]]];
Table[CoefficientList[P[n], x], {n, 0, 9}] (* Jean-François Alcover, Jun 15 2019, from Maple *)

# uses[riordan_square from A321620]
R = riordan_square((1 - 3*x)^(-1/3), 9, True).inverse()
for n in (0..8): print([(-1)^(n-k)*c for (k, c) in enumerate(R.row(n)[:n+1])])

Let R be the inverse of the Riordan square [see A321620] of (1 - 3*x)^(-1/3) then T(n, k) = (-1)^(n-k)*R(n, k).

A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 7, 5, 1, 4, 14, 16, 7, 1, 5, 25, 41, 29, 9, 1, 6, 41, 91, 92, 46, 11, 1, 7, 63, 182, 246, 175, 67, 13, 1, 8, 92, 336, 582, 550, 298, 92, 15, 1, 9, 129, 582, 1254, 1507, 1079, 469, 121, 17, 1, 10, 175, 957, 2508, 3718, 3367, 1925, 696, 154

Offset: 0

Philippe Deléham, Jan 24 2014

Triangle T(n,k), read by rows, given by (1, 1, -1, 1, 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.
Row sums are A111282(n+1) = A025169(n-1).
Diagonal sums are A122391(n+1) = A003945(n-1).

			Triangle begins:
  1;
  1,  1;
  2,  3,   1;
  3,  7,   5,   1;
  4, 14,  16,   7,   1;
  5, 25,  41,  29,   9,  1;
  6, 41,  91,  92,  46, 11,  1;
  7, 63, 182, 246, 175, 67, 13, 1;

Cf. Columns: A028310, A004006.
Cf. Diagonals: A000012, A005408, A130883.
Cf. Similar sequences: A078812, A085478, A111125, A128908, A165253, A207606.
Cf. A321620.

# The function RiordanSquare is defined in A321620.
RiordanSquare(1+x/(1-x)^2, 8); # Peter Luschny, Mar 06 2022

CoefficientList[#, y] & /@
CoefficientList[
Series[(1 - x + x^2)/(1 - 2*x - x*y + x^2), {x, 0, 12}], x] (* Wouter Meeussen, Jan 25 2014 *)

G.f.: (1 - x + x^2)/(1 - 2*x - x*y + x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = 2, T(2,1) = 3, T(2,2) = 1, T(n,k) = 0 if k < 0 or k > n.
The Riordan square (see A321620) of 1 + x/(1 - x)^2. - Peter Luschny, Mar 06 2022

A285072 Triangle read by rows: coefficients of the Laplacian polynomial of the n-path graph P_n.

Original entry on oeis.org

0, -1, 0, -2, 1, 0, -3, 4, -1, 0, -4, 10, -6, 1, 0, -5, 20, -21, 8, -1, 0, -6, 35, -56, 36, -10, 1, 0, -7, 56, -126, 120, -55, 12, -1, 0, -8, 84, -252, 330, -220, 78, -14, 1, 0, -9, 120, -462, 792, -715, 364, -105, 16, -1, 0, -10, 165, -792, 1716, -2002, 1365, -560, 136, -18, 1

Offset: 1

Eric W. Weisstein, Apr 09 2017

Version of A053122 with row-ending 0's and differing signs.

			Table starts:
  0
 -1    0
 -2    1    0
 -3    4   -1     0
 -4   10   -6     1     0
 -5   20  -21     8    -1    0
 -6   35  -56    36   -10    1     0
 -7   56  -126  120   -55   12    -1   0
 -8   84  -252  330  -220   78   -14   1   0
 -9  120  -462  792  -715  364  -105  16  -1   0

Eric Weisstein's World of Mathematics, Path Graph
Eric Weisstein's World of Mathematics, Laplacian Polynomial

Cf. A053122 (version lacking row-ending 0's and with differing signs).

Cf. A321620.

S := proc(n, k) option remember;
if n <= k then 0 elif k = 0 then (-1)^n*n
else S(n-1, k-1) - S(n-2, k) - 2*S(n-1, k) fi end:
T := (n, k) -> (-1)^(n+1)*S(n, k):
seq(seq(T(n, k), k=0..n), n=0..10); # Peter Luschny, Apr 03 2020

CoefficientList[Table[CharacteristicPolynomial[KirchhoffMatrix[PathGraph[Range[n]]], x], {n, 10}], x] // Flatten
CoefficientList[LinearRecurrence[{2 - x, -1}, {-x, (-2 + x) x}, 10], x] // Flatten
CoefficientList[Table[(-1)^(n + 1) x^(1/2) ChebyshevU[2 n - 1, -Sqrt[x]/2], {n, 10}], x] // Flatten
CoefficientList[Table[(2^-n ((2 - Sqrt[-4 + x] Sqrt[x] - x)^n - (2 + Sqrt[-4 + x] Sqrt[x] - x)^n))/Sqrt[(-4 + x)/x], {n, 10}] // Expand // FullSimplify, x] // Flatten
T[n_,k_]:=(-1)^(k+1)*Binomial[n+k,2*k+1];Flatten[Table[T[n,k],{n,0,10},{k,0,n}]] (* Detlef Meya, Oct 09 2023 *)

# uses[riordan_square from A321620]
# Returns the triangle as a matrix.
riordan_square(-x/(1 - x)^2, 9) # Peter Luschny, Apr 03 2020

T(n,k) = (-1)^(k+1)*binomial[n+k,2*k+1]; 0 <= n <= k - Detlef Meya, Oct 09 2023

A321573 Row sums of A321624.

Original entry on oeis.org

1, 2, 8, 22, 68, 202, 608, 1822, 5468, 16402, 49208, 147622, 442868, 1328602, 3985808, 11957422, 35872268, 107616802, 322850408, 968551222, 2905653668, 8716961002, 26150883008, 78452649022, 235357947068, 706073841202, 2118221523608, 6354664570822, 19063993712468

Offset: 0

Peter Luschny, Nov 22 2018

nonn

Index entries for linear recurrences with constant coefficients, signature (2, 3).

Cf. A321624, A321620.

(* The function RiordanSquare is defined in A321620. *)
Lucas = 1 + x*(1 + 2x)/(1 - x - x^2);
M = RiordanSquare[Lucas, 29]; Map[Total, M]

From Philippe Deléham, Feb 05 2020: (Start)
G.f.: (1+x^2)/(1-2x-3*x^2).
a(0)=1, a(1)=2, a(2)=8, a(n)=2*a(n-1)+3*a(n-2) , for n>2.
a(0)=1, a(n) = (5*3^(n-1)-(-1)^(n-1))/2, for n>0 . (End)

A321574 Row sums of A321623.

Original entry on oeis.org

1, 4, 20, 108, 604, 3444, 19876, 115644, 676748, 3977316, 23451700, 138634764, 821221692, 4872734676, 28952250564, 172222729308, 1025461169772, 6110912705220, 36441840179284, 217450403367852, 1298228971572764, 7754336590794804, 46335720234364900

Offset: 0

Peter Luschny, Nov 22 2018

nonn

Cf. A321623, A321620.

(* The function RiordanSquare is defined in A321620. *)
LargeSchröder =  (1 - x - Sqrt[1 - 6x + x^2])/(2x);
M = RiordanSquare[LargeSchröder, 23]; Map[Total, M]
(* Alternative: *)
CoefficientList[Series[1/(Sqrt[1-6x+x^2]-x),{x,0,22}],x] (* Stefano Spezia, Feb 05 2020 *)

G.f.: 1/(sqrt(1 - 6*x + x^2) - x). - Philippe Deléham, Feb 05 2020

A321621 The Riordan square of the Motzkin numbers, triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 8, 5, 1, 9, 21, 18, 7, 1, 21, 55, 58, 32, 9, 1, 51, 145, 177, 123, 50, 11, 1, 127, 385, 525, 431, 224, 72, 13, 1, 323, 1030, 1532, 1429, 889, 369, 98, 15, 1, 835, 2775, 4428, 4572, 3269, 1639, 566, 128, 17, 1

Offset: 0

Peter Luschny, Nov 22 2018

			[0][   1]
[1][   1,    1]
[2][   2,    3,    1]
[3][   4,    8,    5,    1]
[4][   9,   21,   18,    7,    1]
[5][  21,   55,   58,   32,    9,    1]
[6][  51,  145,  177,  123,   50,   11,   1]
[7][ 127,  385,  525,  431,  224,   72,  13,   1]
[8][ 323, 1030, 1532, 1429,  889,  369,  98,  15,  1]
[9][ 835, 2775, 4428, 4572, 3269, 1639, 566, 128, 17, 1]

Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 16, 29.

T(n, 0) = A001006 (Motzkin), A111961 (row sums), A000007 (alternating row sums).

Cf. A321620.

# The function RiordanSquare is defined in A321620.
Motzkin := (1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2); RiordanSquare(Motzkin, 10);

(* The function RiordanSquare is defined in A321620. *)
Motzkin = (1 - x - Sqrt[1 - 2 x - 3 x^2])/(2 x^2);
M = RiordanSquare[Motzkin, 10];
M // Flatten (* Jean-François Alcover, Nov 24 2018 *)

# uses[riordan_square from A321620]
riordan_square((1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2), 10)

A321622 The Riordan square of the Fine numbers, triangle read by rows, T(n, k) for 0 <= k<= n.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 6, 10, 7, 3, 1, 1, 18, 31, 19, 10, 4, 1, 1, 57, 97, 61, 29, 13, 5, 1, 1, 186, 316, 196, 96, 40, 16, 6, 1, 1, 622, 1054, 652, 316, 136, 52, 19, 7, 1, 1, 2120, 3586, 2210, 1072, 458, 181, 65, 22, 8, 1, 1

Offset: 0

Peter Luschny, Nov 22 2018

Fine numbers as defined in A000957 have a(0) = 0 whereas our variant has a(0) = 1. The rows sums of the triangle are |A002420|.

			[0] [    1]
[1] [    1,     1]
[2] [    0,     1,     1]
[3] [    1,     1,     1,     1]
[4] [    2,     4,     2,     1,     1]
[5] [    6,    10,     7,     3,     1,     1]
[6] [   18,    31,    19,    10,     4,     1,   1]
[7] [   57,    97,    61,    29,    13,     5,   1,  1]
[8] [  186,   316,   196,    96,    40,    16,   6,  1,  1]
[9] [  622,  1054,   652,   316,   136,    52,  19,  7,  1,  1]

T(n, 0) = A000957 (Fine), |A002420| (row sums), A000007 (alternating row sums).

Cf. A321620.

# The function RiordanSquare is defined in A321620.
Fine := 1 + (1 - sqrt(1 - 4*x))/(3 - sqrt(1 - 4*x)); RiordanSquare(Fine, 10);

(* The function RiordanSquare is defined in A321620. *)
FineGF = 1 + (1 - Sqrt[1 - 4x])/(3 - Sqrt[1 - 4x]);
RiordanSquare[FineGF, 10] (* Jean-François Alcover, Jun 15 2019, from Maple *)

# uses[riordan_square from A321620]
riordan_square(1 + (1 - sqrt(1 - 4*x))/(3 - sqrt(1 - 4*x)), 10)

cp's OEIS Frontend

A321624 The Riordan square of the Lucas numbers, triangle read by rows, T(n, k) for 0 <= k <= n.

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A321625 The Riordan square of the swinging factorial (A056040), triangle read by rows, T(n, k) for 0 <= k<= n.

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A321627 The Riordan square of the double factorial of odd numbers. Triangle T(n, k), 0 <= k <= n, read by rows.

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A322944 Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.

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A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

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A285072 Triangle read by rows: coefficients of the Laplacian polynomial of the n-path graph P_n.

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A321573 Row sums of A321624.

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A321574 Row sums of A321623.