A202503 -id:A202503 - cp's OEIS frontend

A202453 Fibonacci self-fusion matrix, by antidiagonals.

1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 5, 6, 5, 5, 8, 8, 9, 9, 8, 8, 13, 13, 15, 15, 15, 13, 13, 21, 21, 24, 24, 24, 24, 21, 21, 34, 34, 39, 39, 40, 39, 39, 34, 34, 55, 55, 63, 63, 64, 64, 63, 63, 55, 55, 89, 89, 102, 102, 104, 104, 104, 102, 102, 89, 89, 144, 144, 165, 165

Offset: 1

Clark Kimberling, Dec 19 2011

The Fibonacci self-fusion matrix, F, is the fusion P**Q, where P and Q are the lower and upper triangular Fibonacci matrices.  See A193722 for the definition of fusion of triangular arrays.
Every term F(n,k) of F is a product of two Fibonacci numbers; indeed,
F(n,k)=F(n)*F(k+1) if k is even;
F(n,k)=F(n+1)*F(k) if k is odd.
antidiagonal sums: (1,2,6,12,...), A054454
diagonal (1,2,6,15,...), A001654
diagonal (1,3,9,24,...), A064831
diagonal (2,5,15,39,..), A059840
diagonal (3,8,24,63,..), A080097
diagonal (5,13,39,102,...), A080143
diagonal (8,21,63,165,...), A080144
principal submatrix sums, A202462
All the principal submatrices are invertible, and the terms in the inverses are in {-3,-2,-1,0,1,2,3}.

			Northwest corner:
1...1....2....3....5....8....13
1...2....3....5....8...13....21
2...3....6....9...15...24....39
3...5....9...15...24...39....63
5...8...15...24...40...64...104

Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.

Cf. A000045, A202451, A202452, A202503 (Fibonacci fission array).

n = 12;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
P = Transpose[Q]; F = P.Q;
Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202452 as a sequence *)
Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202451 as a sequence *)
Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202453 as a sequence *)
TableForm[Q]  (* A202451, upper tri. Fibonacci array *)
TableForm[P]  (* A202452, lower tri. Fibonacci array *)
TableForm[F]  (* A202453, Fibonacci fusion array *)
TableForm[FactorInteger[F]]

Matrix product P*Q, where P, Q are the lower and upper triangular Fibonacci matrices, A202451 and A202452.

A129267 Triangle with T(n,k) = T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k) and T(0,0)=1 .

Original entry on oeis.org

1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -1, -3, -2, 1, 1, 0, -2, -5, -3, 1, 1, 1, 2, -2, -7, -4, 1, 1, 1, 5, 7, -1, -9, -5, 1, 1, 0, 3, 12, 15, 1, -11, -6, 1, 1, -1, -3, 3, 21, 26, 4, -13, -7, 1, 1, -1, -7, -15, -3, 31, 40, 8, -15, -8, 1, 1

Offset: 0

Philippe Deléham, Jun 08 2007

Triangle T(n,k), 0<=k<=n, read by rows given by [1,-1,1,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] where DELTA is the operator defined in A084938 . Riordan array (1/(1-x+x^2),(x*(1-x))/(1-x+x^2)); inverse array is (1/(1+x),(x/(1+x))*c(x/(1+x))) where c(x)is g.f. of A000108 .

Row sums are ( with the addition of a first row {0}): 0, 1, 2, 2, 0, -4, -8, -8, 0, 16, 32,... (see A009545). - Roger L. Bagula, Nov 15 2009

			Triangle begins:
   1;
   1,  1;
   0,  1,   1;
  -1, -1,   1,  1;
  -1, -3,  -2,  1,  1;
   0, -2,  -5, -3,  1,   1;
   1,  2,  -2, -7, -4,   1,   1;
   1,  5,   7, -1, -9,  -5,   1,   1;
   0,  3,  12, 15,  1, -11,  -6,   1,  1;
  -1, -3,   3, 21, 26,   4, -13,  -7,  1, 1;
  -1, -7, -15, -3, 31,  40,   8, -15, -8, 1, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A063967, A167925, A199324, A202503, A202551.

T:= proc(n, k) option remember;
      if k<0 or  k>n  then 0
    elif n=0 and k=0 then 1
    else T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 14 2020

m = {{a, 1}, {-1, 1}}; v[0]:= {0, 1}; v[n_]:= v[n] = m.v[n-1]; Table[CoefficientList[v[n][[1]], a], {n, 0, 10}]//Flatten (* Roger L. Bagula, Nov 15 2009 *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0 && k==0, 1, T[n-1, k-1] + T[n-1, k] - T[n-2, k-1] - T[n-2, k] ]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 14 2020 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (n==0 and k==0): return 1
    else: return T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 14 2020

Sum{k=0..n} T(n,k)*x^k = { (-1)^n*A057093(n), (-1)^n*A057092(n), (-1)^n*A057091(n), (-1)^n*A057090(n), (-1)^n*A057089(n), (-1)^n*A057088(n), (-1)^n*A057087(n), (-1)^n*A030195(n+1), (-1)^n*A002605(n), A039834(n+1), A000007(n), A010892(n), A099087(n), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n), A057086(n) } for x=-11, -10, ..., 8, 9, respectively .
Sum{k=0..n} T(n,k)*A000045(k) = A100334(n).
Sum{k=0..floor(n/2)} T(n-k,k) = A050935(n+2).
T(n,k)= Sum{j>=0} A109466(n,j)*binomial(j,k).
T(n,k) = (-1)^(n-k)*A199324(n,k) = (-1)^k*A202551(n,k) = A202503(n,n-k). - Philippe Deléham, Mar 26 2013
G.f.: 1/(1-x*y+x^2*y-x+x^2). - R. J. Mathar, Aug 11 2015

Riordan array definition corrected by Ralf Stephan, Jan 02 2014

A193917 Triangular array: the self-fusion of (p(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 3, 6, 9, 3, 5, 9, 15, 24, 5, 8, 15, 24, 40, 64, 8, 13, 24, 39, 64, 104, 168, 13, 21, 39, 63, 104, 168, 273, 441, 21, 34, 63, 102, 168, 272, 441, 714, 1155, 34, 55, 102, 165, 272, 440, 714, 1155, 1870, 3025, 55, 89, 165, 267, 440, 712, 1155

Offset: 0

Clark Kimberling, Aug 09 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.  (Fusion is defined at A193822; fission, at A193742; see A202503 and A202453 for infinite-matrix representations of fusion and fission.)
First five rows of P (triangle of coefficients of polynomials p(n,x)):
1
1...1
1...1...2
1...1...2...3
1...1...2...3...5
First eight rows of A193917:
1
1...1
1...2...3
2...3...6...9
3...5...9...15...24
5...8...15..24...40...64
8...13..24..39...64...104..168
13..21..39..63...104..168..273..441
...
col 1:  A000045
col 2:  A000045
col 3:  A022086
col 4:  A022086
col 5:  A022091
col 6:  A022091
col 7:  A022355
col 8:  A022355
right edge, w(n,n):  A064831
w(n,n-1):  A001654
w(n,n-2):  A064831
w(n,n-3):  A059840
w(n,n-4):  A080097
w(n,n-5):  A080143
w(n,n-6):  A080144
Suppose n is an even positive integer and w(n+1,x) is the polynomial matched to row n+1 of A193917 as in the Mathematica program (and definition of fusion at A193722), where the first row is counted as row 0.

			First six rows:
1
1...1
1...2...3
2...3...6....9
3...5...9....15...24
5...8...15...24...40...64

Cf. A193722, A064831, A193918, A194000, A194001.

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := p[n, x];
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}]]  (* A193917 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193918 *)

A194000 Triangular array: the self-fission of (p(n,x)), where sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 2, 3, 3, 5, 9, 5, 8, 15, 24, 8, 13, 24, 39, 64, 13, 21, 39, 63, 104, 168, 21, 34, 63, 102, 168, 272, 441, 34, 55, 102, 165, 272, 440, 714, 1155, 55, 89, 165, 267, 440, 712, 1155, 1869, 3025, 89, 144, 267, 432, 712, 1152, 1869, 3024, 4895, 7920, 144, 233

Offset: 0

Clark Kimberling, Aug 11 2011

See A193917 for the self-fusion of the same sequence of polynomials.  (Fusion is defined at A193822; fission, at A193842; see A202503 and A202453 for infinite-matrix representations of fusion and fission.)
...
First five rows of P (triangle of coefficients of polynomials p(n,x)):
1
1...1
1...1...2
1...1...2...3
1...1...2...3...5
First eight rows of A194000:
1
2....3
3....5....9
5....8....15...24
8....13...24...39...64
13...21...29...63...104...168
21...34...63...102..168...272...441
34...55...102..165..272...440...714..1155
...
col 1:  A000045
col 2:  A000045
col 3:  A022086
col 4:  A022086
col 5:  A022091
col 6:  A022091
right edge, d(n,n):  A064831
d(n,n-1):  A059840
d(n,n-2):  A080097
d(n,n-3):  A080143
d(n,n-4):  A080144
...
Suppose n is an odd positive integer and d(n+1,x) is the polynomial matched to row n+1 of A194000 as in the Mathematica program (and definition of fission at A193842), where the first row is counted as row 0.

			First six rows:
1
2....3
3....5....9
5....8....15...24
8....13...24...39...64
13...21...29...63...104...168
...
Referring to the matrix product for fission at A193842,
the row (5,8,15,24) is the product of P(4) and QQ, where
P(4)=(p(4,4), p(4,3), p(4,2), p(4,1))=(5,3,2,1); and
QQ is the 4x4 matrix
(1..1..2..3)
(0..1..1..2)
(0..0..1..1)
(0..0..0..1).

Cf. A193842, A194001, A193917, A193918.

z = 11;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := p[n, x];
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A194000 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A194001 *)

A202502 Modified lower triangular Fibonacci matrix, by antidiagonals.

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 2, 5, 0, 0, 1, 3, 8, 0, 0, 0, 2, 5, 13, 0, 0, 0, 1, 3, 8, 21, 0, 0, 0, 0, 2, 5, 13, 34, 0, 0, 0, 0, 1, 3, 8, 21, 55, 0, 0, 0, 0, 0, 2, 5, 13, 34, 89, 0, 0, 0, 0, 0, 1, 3, 8, 21, 55, 144, 0, 0, 0, 0, 0, 0, 2, 5, 13, 34, 89, 233, 0, 0, 0, 0, 0, 0, 1, 3, 8, 21, 55

Offset: 1

Clark Kimberling, Dec 20 2011

This matrix, P, is used to form the Fibonacci self-fission matrix as the product P*Q, where Q is the upper triangular Fibonacci matrix, A202451. To form P, delete the main diagonal of the transpose of Q.

			Northwest corner:
1...0...0...0...0...0...0...0...0
2...1...0...0...0...0...0...0...0
3...2...1...0...0...0...0...0...0
5...3...2...1...1...0...0...0...0
8...5...3...2...1...1...0...0...0

Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.

Cf. A202503, A202451, A202452, A202453, A000045.

n = 14;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
Qt = Transpose[Q]; P1 = Qt - IdentityMatrix[n];
P = P1[[Range[2, n], Range[1, n]]];
F = P.Q;
Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202502 as a sequence *)
Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202451 as a sequence *)
Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202503 as a sequence *)
TableForm[P]  (* A202502, modified lower triangular Fibonacci matrix *)
TableForm[Q] (* A202451, upper tri. Fibonacci matrix *)
TableForm[F] (* A202503, Fibonacci self-fission matrix *)

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A202453 Fibonacci self-fusion matrix, by antidiagonals.

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A129267 Triangle with T(n,k) = T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k) and T(0,0)=1 .

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A193917 Triangular array: the self-fusion of (p(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

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A194000 Triangular array: the self-fission of (p(n,x)), where sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

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A202502 Modified lower triangular Fibonacci matrix, by antidiagonals.

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