A193737 Mirror of the triangle A193736.

1, 1, 1, 1, 2, 1, 2, 4, 3, 1, 3, 8, 8, 4, 1, 5, 15, 19, 13, 5, 1, 8, 28, 42, 36, 19, 6, 1, 13, 51, 89, 91, 60, 26, 7, 1, 21, 92, 182, 216, 170, 92, 34, 8, 1, 34, 164, 363, 489, 446, 288, 133, 43, 9, 1, 55, 290, 709, 1068, 1105, 826, 455, 184, 53, 10, 1, 89, 509, 1362, 2266, 2619, 2219, 1414, 682, 246, 64, 11, 1

Offset

0,5

Comments

This triangle is obtained by reversing the rows of the triangle A193736.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

Write w(n,k) for the triangle at A193736. This is then given by w(n,n-k).

T(0,0) = T(1,0) = T(1,1) = T(2,0) = 1; T(n,k) = 0 if k<0 or k>n; T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k). - Philippe Deléham, Feb 13 2020

From G. C. Greubel, Oct 24 2023: (Start)

T(n, 0) = Fibonacci(n) + [n=0] = A324969(n+1).

T(n, n-1) = n, for n >= 1.

T(n, n-2) = A034856(n-1), for n >= 2.

T(2*n, n) = A330793(n).

Sum_{k=0..n} T(n,k) = A052542(n).

Sum_{k=0..n} (-1)^k * T(n,k) = A000007(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A011782(n).

Sum_{k=0..floor(n/2)} (-1)^k * T(n-k,k) = A019590(n). (End)

Example

First six rows:
  1;
  1,  1;
  1,  2,  1;
  2,  4,  3,  1;
  3,  8,  8,  4,  1;
  5, 15, 19, 13,  5,  1;

Mathematica

(* First program *)
z=20;
p[0, x_]:= 1;
p[n_, x_]:= Fibonacci[n+1, x] /; n > 0
q[n_, x_]:= (x + 1)^n;
t[n_, k_]:= Coefficient[p[n, x], x^(n-k)];
t[n_, n_]:= p[n, x] /. x -> 0;
w[n_, x_]:= Sum[t[n, k]*q[n-k+1, x], {k, 0, n}]; w[-1, x_] := 1;
g[n_]:= CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193736 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]          (* A193737 *)
(* Additional programs *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[1 + 1/(1 - x - x^2), 11]//Flatten  (* Peter Luschny, Feb 27 2021 *)
T[n_, k_]:= T[n, k]= If[n<3, Binomial[n, k], T[n-1, k] + T[n-1, k-1] + T[n-2, k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 24 2023 *)

Prog

(Magma)
function T(n, k) // T = A193737
  if k lt 0 or n lt 0 then return 0;
  elif n lt 3 then return Binomial(n, k);
  else return T(n - 1, k) + T(n - 1, k - 1) + T(n - 2, k);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 24 2023
(SageMath)
def T(n, k): # T = A193737
    if (n<3): return binomial(n, k)
    else: return T(n-1, k) +T(n-1, k-1) +T(n-2, k)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023

Crossrefs

Cf. A000007, A011782 (diagonal sums), A019590, A052542 (row sums).

Cf. A034856, A193736, A321620, A324969, A330793.

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 04 2011

Status

approved

A091173 Triangle, read by rows, where the n-th row lists the coefficients of the polynomial of degree n, with root -1, that generates the n-th diagonal of this sequence.

1, 1, 1, 1, 2, 1, 2, 4, 3, 1, 4, 10, 9, 4, 1, 10, 28, 30, 16, 5, 1, 30, 90, 108, 68, 25, 6, 1, 106, 328, 426, 304, 130, 36, 7, 1, 420, 1338, 1842, 1444, 700, 222, 49, 8, 1, 1818, 6024, 8706, 7320, 3930, 1404, 350, 64, 9, 1, 8530, 29626, 44736, 39700, 23110, 9150, 2548

Offset

0,5

Comments

The leftmost column (A091174) is determined by the condition that the root of each row polynomial is -1. The next column is T(n,1)=A091175(n+1) (n>=0).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1034

Formula

T(n+k, k) = Sum_{j=0..n} T(n, j) * k^j, with T(0,0)=1, T(0,n)=1 and T(n,0) = -Sum_{j=1..n} T(n, j) * (-1)^j.

Example

For n=3, k=2, T(n+k,k) = T(5,2) = 30 = (2) + (4)2 + (3)2^2 + (1)2^3.
For n=4, k=3, T(n+k,k) = T(7,3) = 304 = (4) + (10)3 + (9)3^2 + (4)3^3 + (1)3^4.
Rows begin with n=0:
1;
1, 1;
1, 2, 1;
2, 4, 3, 1;
4, 10, 9, 4, 1;
10, 28, 30, 16, 5, 1;
30, 90, 108, 68, 25, 6, 1;
106, 328, 426, 304, 130, 36, 7, 1;
420, 1338, 1842, 1444, 700, 222, 49, 8, 1;
1818, 6024, 8706, 7320, 3930, 1404, 350, 64, 9, 1;
8530, 29626, 44736, 39700, 23110, 9150, 2548, 520, 81, 10, 1;
43430, 158012, 248466, 230424, 142890, 61680, 18970, 4288, 738, 100, 11, 1;
240208, 909010, 1483398, 1429236, 931500, 431646, 144858, 35976, 6804, 1010, 121, 12, 1; ...

Mathematica

T[0, _] = 1; T[n_, 0] := T[n, 0] = -Sum[T[n, j]*(-1)^j, {j, 1, n}]; T[n_, k_] := T[n, k] = Sum[T[n-k, j]*k^j, {j, 0, n-k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 21 2015 *)

Prog

(PARI) {T(n, k)=if(n==k, 1, if(n>k&k>0, sum(j=0, n-k, T(n-k, j)*k^j), if(k==0, -sum(j=1, n, T(n, j)*(-1)^j))))}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A091174, A091175.

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 25 2003

Status

approved

A208058 Triangle by rows relating to the factorials, generated from A002260.

1, 1, 1, 1, 2, 1, 2, 4, 3, 1, 6, 12, 9, 4, 1, 24, 48, 36, 16, 5, 1, 120, 240, 180, 80, 25, 6, 1, 720, 1440, 1080, 480, 150, 36, 7, 1, 5040, 10080, 7560, 3360, 1050, 252, 49, 8, 1, 40320, 80640, 60480, 26880, 8400, 2016, 392, 64, 9, 1

Offset

0,5

Comments

Row sums = A054091: (1, 2, 4, 10, 32, 130, 652,...)

Left border = the factorials, A000142 prefaced with a 1.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

Inverse of:

1;

-1, 1;

1, -2, 1;

-1, 2, -3, 1;

1, -2, 3, -4, 1;

..., where triangle A002260 = (1; 1,2; 1,2,3;...)

Example

First few rows of the triangle =
1;
1, 1;
1, 2, 1;
2, 4, 3, 1;
6, 12, 9, 4, 1;
24, 48, 36, 16, 5, 1;
120, 240, 180, 80, 25, 6, 1;
720, 1440, 1080, 480, 150, 36, 7, 1;
5040, 10080, 7560, 3360, 1050, 252, 49, 8, 1;
...

Maple

T:= proc(n) option remember; local M, k;
      M:= Matrix(n+1, (i, j)->
                 `if`(i=j, 1, `if`(i>j, j*(-1)^(i+j), 0)))^(-1);
      seq(M[n+1, k], k=1..n+1)
    end:
seq(T(n), n=0..14);  # Alois P. Heinz, Feb 24 2012

Mathematica

T[n_] := T[n] = Module[{M}, M = Table[If[i == j, 1, If[i>j, j*(-1)^(i+j), 0]], {i, 1, n+1}, {j, 1, n+1}] // Inverse; M[[n+1]]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

Crossrefs

Cf. A000142, A002260, A054091.

Keyword

nonn,tabl

Author

Gary W. Adamson, Feb 22 2012

Status

approved

A101897 Triangle T, read by rows, such that column k equals column 0 of T^(k+1), where column 0 of T allows the n-th row sums to be zero for n>0 and where T^k is the k-th power of T as a lower triangular matrix.

1, -1, 1, 1, -2, 1, -2, 4, -3, 1, 5, -11, 9, -4, 1, -17, 38, -33, 16, -5, 1, 71, -162, 145, -74, 25, -6, 1, -357, 824, -753, 396, -140, 36, -7, 1, 2101, -4892, 4535, -2434, 885, -237, 49, -8, 1, -14203, 33286, -31185, 16982, -6295, 1730, -371, 64, -9, 1, 108609, -255824, 241621, -133012, 50001, -13992, 3073, -548, 81

Offset

0,5

Comments

Column 0 forms A101900. Absolute row sums form A101901.

Links

Table of n, a(n) for n=0..63.

J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.

Formula

T(n, k) = Sum_{j=0..n-k} T(n-k, j)*T(j+k-1, k-1) for n >= k > 0 with T(0, 0) = 1 and T(n, 0) = -Sum_{j=1, n} T(n, j) for n > 0.

Example

Rows begin:
        1;
       -1,     1;
        1,    -2,      1;
       -2,     4,     -3,     1;
        5,   -11,      9,    -4,     1;
      -17,    38,    -33,    16,    -5,    1;
       71,  -162,    145,   -74,    25,   -6,   1;
     -357,   824,   -753,   396,  -140,   36,  -7,   1,
     2101, -4892,   4535, -2434,   885, -237,  49,  -8,  1;
   -14203, 33286, -31185, 16982, -6295, 1730, -371, 64, -9, 1;
      ...

Mathematica

t[n_, k_] := t[n, k] = If[k>n || n<0 || k<0, 0, If[k==n, 1, If[k==0, -Sum[t[n, j], {j, 1, n}], Sum[t[n-k, j]*t[j+k-1, k-1], {j, 0, n-k}]]]]; Table[t[n , k], {n, 0, 10}, {k, 0, n}] //Flatten (* Amiram Eldar, Nov 26 2018 *)

Prog

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

Crossrefs

Cf. A091351, A101900, A101901, A101898, A101899.

Keyword

sign,tabl

Author

Paul D. Hanna, Dec 20 2004

Status

approved

A179750 Triangle T(n,k) read by rows. Matrix inverse of A179749.

1, -1, 1, 1, -2, 1, -2, 4, -3, 1, 4, -8, 7, -4, 1, -7, 14, -14, 11, -5, 1, 11, -22, 25, -25, 16, -6, 1, -18, 36, -44, 51, -41, 22, -7, 1, 35, -70, 83, -99, 92, -63, 29, -8, 1, -76, 152, -166, 188, -190, 155, -92, 37, -9, 1, 166, -332, 337, -354, 373, -345, 247, -129, 46

Offset

1,5

Comments

First column of this triangle is A127926.

Links

Table of n, a(n) for n=1..64.

Example

Table begins:
1,
-1,1,
1,-2,1,
-2,4,-3,1,
4,-8,7,-4,1,
-7,14,-14,11,-5,1,
11,-22,25,-25,16,-6,1,
-18,36,-44,51,-41,22,-7,1,
35,-70,83,-99,92,-63,29,-8,1,
-76,152,-166,188,-190,155,-92,37,-9,1,
166,-332,337,-354,373,-345,247,-129,46,-10,1,
-358,716,-693,678,-717,719,-592,376,-175,56,-11,1,

Crossrefs

Cf. A179748, A179749, A127926.

Keyword

sign,tabl

Author

Mats Granvik, Jul 26 2010

Status

approved