A193722 -id:A193722 - cp's OEIS frontend

A193822 Mirror of the triangle A193821.

1, 1, 1, 3, 3, 2, 9, 9, 8, 4, 27, 27, 26, 20, 8, 81, 81, 80, 72, 48, 16, 243, 243, 242, 232, 192, 112, 32, 729, 729, 728, 716, 656, 496, 256, 64, 2187, 2187, 2186, 2172, 2088, 1808, 1248, 576, 128, 6561, 6561, 6560, 6544, 6432, 5984, 4864, 3072, 1280, 256

Offset: 0

Clark Kimberling, Aug 06 2011

A193822 is obtained by reversing the rows of the triangle A193821.

			First six rows:
1
1....1
3....3....2
9....9....8.....4
27...27...26....20...8
81...81...80....72...48...16

Cf. A193722, A193821.

p[n_, x_] := (a*x + b)^n
q[0, x_] := 1
q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, 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}]]   (* A193821 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193822 *)

Write w(n,k) for the triangle at A193821. The triangle at A193822 is then given by w(n,n-k).

A193823 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 5, 9, 9, 1, 7, 19, 27, 27, 1, 9, 33, 65, 81, 81, 1, 11, 51, 131, 211, 243, 243, 1, 13, 73, 233, 473, 665, 729, 729, 1, 15, 99, 379, 939, 1611, 2059, 2187, 2187, 1, 17, 129, 577, 1697, 3489, 5281, 6305, 6561, 6561, 1, 19, 163, 835, 2851

Offset: 0

Clark Kimberling, Aug 06 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

			First six rows:
1
1....1
1....3....3
1....5....9....9
1....7....19...27...27
1....9....33...65...81...81

Cf. A193722, A193804. A119258, A193860.

z = 10; a = 2; b = 1;
p[n_, x_] := (a*x + b)^n
q[0, x_] := 1
q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, 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}]]   (* A193823 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193824 *)

From Peter Bala, Jul 16 2013: (Start)
T(n,k) = sum {i = 0..k} binomial(n-1,k-i)*2^(k-i) for 0 <= k <= n.
O.g.f.: (1 - 2*x*t)^2/( (1 - 3*x*t)*(1 - (2*x + 1)*t) ) = 1 + (1 + x)*t + (1 + 3*x + 3*x^2)*t^2 + .... Cf. A193860.
For n >= 1, the n-th row polynomial R(n,x) = 1/(x-1)*( 3^(n-1)*x^(n+1) - (2*x + 1)^(n-1) ). (End)

A193908 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+2)x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=2xq(n-1,x)+1 with q(0,x)=1.

Original entry on oeis.org

1, 2, 1, 8, 6, 3, 24, 20, 12, 6, 80, 64, 40, 22, 11, 256, 208, 128, 72, 38, 19, 832, 672, 416, 232, 124, 64, 32, 2688, 2176, 1344, 752, 400, 208, 106, 53, 8704, 7040, 4352, 2432, 1296, 672, 344, 174, 87, 28160, 22784, 14080, 7872, 4192, 2176, 1112, 564

Offset: 0

Clark Kimberling, Aug 09 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
First five rows of P (triangle of coefficients of polynomials p(n,x)):
1
1...2
1...2...3
1...2...3...5
1...2...3...5...8
First five rows of Q:
1
2...1
4...2..1
8...4...2...1
16..8...4...2...1

			First six rows:
1
2....1
8....6....3
24...20...12...6
80...64...40...22...11
256..208..128..72...38...19

Cf. A193722, A193909.

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 2]*x^(n - k), {k, 0, n}];
q[n_, x_] := 2 x*q[n - 1, x] + 1 ; q[0, x_] := 1;
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, 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}]]  (* A193908 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193909 *)

A196664 Expansion of g.f. (1-5x)/(1-16x).

Original entry on oeis.org

1, 11, 176, 2816, 45056, 720896, 11534336, 184549376, 2952790016, 47244640256, 755914244096, 12094627905536, 193514046488576, 3096224743817216, 49539595901075456, 792633534417207296, 12682136550675316736, 202914184810805067776, 3246626956972881084416

Offset: 0

Philippe Deléham, Oct 05 2011

Index entries for linear recurrences with constant coefficients, signature (16).

Cf. A001025, A193722, A196661, A196662, A196663.

Join[{1},Table[11 16^(n-1),{n,20}]] (* Harvey P. Dale, Oct 20 2011 *)

a(0) = 1, a(n) = 11*16^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*5^k.
From Elmo R. Oliveira, Mar 18 2025: (Start)
E.g.f.: (11*exp(16*x) + 5)/16.
a(n) = 16*a(n-1). (End)

A202670 Symmetric matrix based on A000290 (the squares), by antidiagonals.

Original entry on oeis.org

1, 4, 4, 9, 17, 9, 16, 40, 40, 16, 25, 73, 98, 73, 25, 36, 116, 184, 184, 116, 36, 49, 169, 298, 354, 298, 169, 49, 64, 232, 440, 584, 584, 440, 232, 64, 81, 305, 610, 874, 979, 874, 610, 305, 81, 100, 388, 808, 1224, 1484, 1484, 1224, 808, 388, 100, 121

Offset: 1

Clark Kimberling, Dec 22 2011

Let s=(1,4,9,16,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s.  Let T' be the transpose of T.  Then A202670 represents the matrix product M=T'*T.  M is the self-fusion matrix of s, as defined at A193722.  See A202671 for characteristic polynomials of principal submatrices of M.
...
row 1 (1,4,9,16,...) A000290
row 2 (4,17,40,73,...) A145995
diagonal (1,17,98,354,...) A000538
antidiagonal sums (1,8,35,112,...) A040977
...
The n-th "square border sum" m(n,1)+m(n,2)+...+m(n,n)+m(n-1,n)+m(n-2,n)+...+m(1,n) is a squared square pyramidal number: [n*(n+1)*(2*n+1)/6]^2; see A000330.

			Northwest corner:
1.....4......9....16....25
4....17.....40....73...116
9....40.....98...184...298
16...73....184...354...584
25...116...298...584...979

Cf. A000290, A202671, A193722.

U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[ Table[k^2, {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

A202876 Symmetric matrix based on A000071, by antidiagonals.

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 7, 10, 10, 7, 12, 18, 21, 18, 12, 20, 31, 38, 38, 31, 20, 33, 52, 66, 70, 66, 52, 33, 54, 86, 111, 122, 122, 111, 86, 54, 88, 141, 184, 206, 214, 206, 184, 141, 88, 143, 230, 302, 342, 362, 362, 342, 302, 230, 143, 232, 374, 493, 562, 602

Offset: 1

Clark Kimberling, Dec 26 2011

Let s=A000071 (Fibonacci numbers -1), and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202876 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202877 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1....2....4....7....12....20
2....5....10...18...31....52
4....10...21...38...66....111
7....18...38...70...122...206
12...31...66...122..214...362

Cf. A202877, A202876.

s[k_] := -1 + Fibonacci[k + 2];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* A001924 *)
Table[m[1, j], {j, 1, 12}]     (* A000071 *)
Table[m[j, j], {j, 1, 12}]     (* A202462 *)

A203949 Symmetric matrix based on (1,1,0,1,1,0,1,1,0,...), by antidiagonals.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 1, 2, 0, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 0, 2, 1, 1, 4, 1, 1, 2, 0, 1, 1, 1, 3, 2, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 4, 2, 2, 2, 1, 1, 0, 2, 1, 1, 4, 2, 2, 4, 1, 1, 2, 0, 1, 1, 1, 3, 2, 2, 5, 2, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 4, 3, 3

Offset: 1

Clark Kimberling, Jan 08 2012

Let s be the periodic sequence (1,1,0,1,1,0,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A203949 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203950 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1 1 0 1 1 0 1 1 0 1
1 2 1 1 2 1 1 2 1 1
0 1 2 1 1 2 1 1 2 1
1 1 1 3 2 1 3 2 1 3
1 2 1 2 4 2 2 4 2 2
0 1 2 1 2 4 2 2 4 2
1 1 1 3 2 2 5 3 2 5

Cf. A203950, A202453.

t = {1, 1, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t, t}];
s[k_] := t1[[k]];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
   Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M] (* A203949 *)
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

A203951 Symmetric matrix based on (1,0,0,0,1,0,0,0,...), by antidiagonals.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling, Jan 08 2012

Let s be the periodic sequence (1,0,0,0,1,0,0,0,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A203951 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203952 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1 0 0 0 1 0 0 0 1 0
0 1 0 0 0 1 0 0 0 1
0 0 1 0 0 0 1 0 0 0
0 0 0 1 0 0 0 1 0 0
1 0 0 0 2 0 0 0 2 0
0 1 0 0 0 2 0 0 0 2
0 0 1 0 0 0 2 0 0 0
0 0 0 1 0 0 0 2 0 0
1 0 0 0 2 0 0 0 3 0

Cf. A203951, A202453.

t = {1, 0, 0, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t}];
f[k_] := t1[[k]];
U[n_] :=
  NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
   Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
p[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]  (* A203952 *)

A193724 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+1)^n.

Original entry on oeis.org

1, 1, 1, 2, 5, 3, 4, 16, 21, 9, 8, 44, 90, 81, 27, 16, 112, 312, 432, 297, 81, 32, 272, 960, 1800, 1890, 1053, 243, 64, 640, 2736, 6480, 9180, 7776, 3645, 729, 128, 1472, 7392, 21168, 37800, 43092, 30618, 12393, 2187, 256, 3328, 19200, 64512, 139104

Offset: 0

Clark Kimberling, Aug 04 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Triangle T(n,k), read by rows, given by [1,1,0,0,0,0,0,0,0,...] DELTA [1,2,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 04 2011

			First six rows:
   1;
   1,   1;
   2,   5,   3;
   4,  16,  21,   9;
   8,  44,  90,  81,  27;
  16, 112, 312, 432, 297, 81;

Cf. A084938, A193722, A193725.

z = 8; a = 1; b = 2; c = 1; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, 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}]] (* A193724 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193725 *)

T(n,k) = 3*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,0)=T(1,1)=1. - Philippe Deléham, Oct 05 2011

G.f.: (-1+x+2*x*y)/(-1+2*x+3*x*y). - R. J. Mathar, Aug 11 2015

A193726 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^n.

Original entry on oeis.org

1, 1, 2, 2, 9, 10, 4, 28, 65, 50, 8, 76, 270, 425, 250, 16, 192, 920, 2200, 2625, 1250, 32, 464, 2800, 9000, 16250, 15625, 6250, 64, 1088, 7920, 32000, 77500, 112500, 90625, 31250, 128, 2496, 21280, 103600, 315000, 612500, 743750, 515625, 156250

Offset: 0

Clark Kimberling, Aug 04 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (2,3,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

			First six rows:
   1;
   1,   2;
   2,   9,  10;
   4,  28,  65,   50;
   8,  76, 270,  425,  250;
  16, 192, 920, 2200, 2625, 1250;

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

Cf. A002532, A011782, A020699, A045873, A081040, A084938.

Cf. A133494, A141413, A169634, A193722, A193727.

function T(n, k) // T = A193726
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return k+1;
  else return 2*T(n-1, k) + 5*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 02 2023

(* First program *)
z = 8; a = 1; b = 2; c = 1; d = 2;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, 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}]]  (* A193726 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193727 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, k+1, 2*T[n-1, k] + 5*T[n -1, k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 02 2023 *)

def T(n, k): # T = A193726
    if (k<0 or k>n): return 0
    elif (n<2): return k+1
    else: return 2*T(n-1, k) + 5*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 02 2023

T(n,k) = 5*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-x-3*x*y)/(1-2*x-5*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 02 2023: (Start)
T(n, 0) = A011782(n).
T(n, n) = A020699(n).
T(n, n-1) = A081040(n-1).
Sum_{k=0..n} T(n, k) = A169634(n-1) + (4/7)*[n=0].
Sum_{k=0..n} (-1)^k * T(n, k) = (-1)^n*A133494(n) = -A141413(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A002532(n) + 2*A002532(n-1) + (3/5)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A045873(n) - 2*A045873(n-1) + (3/5)*[n=0]. (End)

cp's OEIS Frontend

A193822 Mirror of the triangle A193821.

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A193823 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1.

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A193908 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+2)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=2x*q(n-1,x)+1 with q(0,x)=1.

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A196664 Expansion of g.f. (1-5*x)/(1-16*x).

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A202670 Symmetric matrix based on A000290 (the squares), by antidiagonals.

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A202876 Symmetric matrix based on A000071, by antidiagonals.

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A203949 Symmetric matrix based on (1,1,0,1,1,0,1,1,0,...), by antidiagonals.

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A203951 Symmetric matrix based on (1,0,0,0,1,0,0,0,...), by antidiagonals.

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A193724 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+1)^n.

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A193908 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+2)x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=2xq(n-1,x)+1 with q(0,x)=1.

A196664 Expansion of g.f. (1-5x)/(1-16x).