A115262 -id:A115262 - cp's OEIS frontend

A202605 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the Fibonacci self-fusion matrix (A202453).

1, -1, 1, -3, 1, 1, -6, 9, -1, 1, -9, 26, -24, 1, 1, -12, 52, -96, 64, -1, 1, -15, 87, -243, 326, -168, 1, 1, -18, 131, -492, 1003, -1050, 441, -1, 1, -21, 184, -870, 2392, -3816, 3265, -1155, 1, 1, -24, 246, -1404, 4871, -10500, 13710

Offset: 1

Clark Kimberling, Dec 21 2011

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are positive and interlace the zeros of p(n+1). (See the references and examples.)
Following is a guide to sequences (f(n)) for symmetric matrices (self-fusion matrices) and characteristic polynomials. Notation: F(k)=A000045(k) (Fibonacci numbers); floor(n*tau)=A000201(n) (lower Wythoff sequence); "periodic x,y" represents the sequence (x,y,x,y,x,y,...).
f(n)........ symmetric matrix.. char. polynomial
1............... A087062....... A202672
n............... A115262....... A202673
n^2............. A202670....... A202671
2n-1............ A202674....... A202675
3n-2............ A202676....... A202677
n(n+1)/2........ A185957....... A202678
2^n-1........... A202873....... A202767
2^(n-1)......... A115216....... A202868
floor(n*tau).... A202869....... A202870
F(n)............ A202453....... A202605
F(n+1).......... A202874....... A202875
Lucas(n)........ A202871....... A202872
F(n+2)-1........ A202876....... A202877
F(n+3)-2........ A202970....... A202971
(F(n))^2........ A203001....... A203002
(F(n+1))^2...... A203003....... A203004
C(2n,n)......... A115255....... A203005
(-1)^(n+1)...... A003983....... A076757
periodic 1,0.... A203905....... A203906
periodic 1,0,0.. A203945....... A203946
periodic 1,0,1.. A203947....... A203948
periodic 1,1,0.. A203949....... A203950
periodic 1,0,0,0 A203951....... A203952
periodic 1,2.... A203953....... A203954
periodic 1,2,3.. A203955....... A203956
...
In the cases listed above, the zeros of the characteristic polynomials are positive. If more general symmetric matrices are used, the zeros are all real but not necessarily positive - but they do have the interlace property. For a guide to such matrices and polynomials, see A202605.

			The 1st principal submatrix (ps) of A202453 is {{1}} (using Mathematica matrix notation), with p(1) = 1-x and zero-set {1}.
...
The 2nd ps is {{1,1},{1,2}}, with p(2) = 1-3x+x^2 and zero-set {0.382..., 2.618...}.
...
The 3rd ps is {{1,1,2},{1,2,3},{2,3,6}}, with p(3) = 1-6x+9x^2-x^3 and zero-set {0.283..., 0.426..., 8.290...}.
  ...
Top of the array A202605:
  1,   -1;
  1,   -3,    1;
  1,   -6,    9,   -1;
  1,   -9,   26,  -24,    1;
  1,  -12,   52,  -96,   64,   -1;
  1,  -15,   87, -243,  326, -168,    1;

S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.
Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.
A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Cf. A000045, A202453.

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

A185957 Second accumulation array of the array min{n,k}, by antidiagonals.

Original entry on oeis.org

1, 3, 3, 6, 10, 6, 10, 21, 21, 10, 15, 36, 46, 36, 15, 21, 55, 81, 81, 55, 21, 28, 78, 126, 146, 126, 78, 28, 36, 105, 181, 231, 231, 181, 105, 36, 45, 136, 246, 336, 371, 336, 246, 136, 45, 55, 171, 321, 461, 546, 546, 461, 321, 171, 55, 66, 210, 406, 606, 756, 812, 756

Offset: 1

Clark Kimberling, Feb 07 2011

A member of the accumulation chain
... < A003982 < A003783 < A115262 < A185957 <...,
where A003783(n,k)=min{n,k}.  See A144112 for the definition of accumulation array.
A185957 also gives the symmetric matrix based on the triangular numbers s=(1,3,6,10,15,....; viz, 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 A185957 represents the matrix product M=T'*T.  M is the self-fusion matrix of s, as defined at A193722.  See A202678 for characteristic polynomials of principal submatrices of M.

			Northwest corner:
1....3....6....10...15
3....10...21...36...55
6....21...46...81...126
10...36...81...146..231

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A003783, A115262.

U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k (k + 1)/2, {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}] (* A000292 *)
Table[m[1, j], {j, 1, 12}] (* A000217 *)
Table[m[2, j], {j, 1, 12}] (* A014105 *)
Table[m[j, j], {j, 1, 12}] (* A024166 *)
Table[m[j, j + 1], {j, 1, 12}] (* A112851 *)
Table[Sum[m[i, n + 1 - i], {i, 1, n}], {n, 1, 12}] (* A001769 *)

A003982 Table read by rows: 1 if x = y, 0 otherwise, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Marc LeBrun

Also called the delta function.
From Clark Kimberling, Feb 07 2011: (Start)
In rectangular format, the infinite identity matrix and the weight array of A003783(n,k)=min{n,k}; in the accumulation chain
... < A003982 < A003783 < A115262 < A185957 < ... . See A144112 for definitions of weight array and accumulation array. (End)

			Table begins
  1;
  0, 0;
  0, 1, 0;
  0, 0, 0, 0;
  0, 0, 1, 0, 0;
  ....
Northwest corner when formatted as a rectangular array:
  1 0 0 0 0 0 0 0
  0 1 0 0 0 0 0 0
  0 0 1 0 0 0 0 0
  0 0 0 1 0 0 0 0
  0 0 0 0 1 0 0 0

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for characteristic functions

Characteristic function of A001844. Antidiagonal sums and main diagonal is A000012.

Cf. also A286100.

f[n_,k_]:=0; f[n_,n_]:=1;
TableForm[Table[f[n,k],{n,1,10},{k,1,10}]] (* array *)
Table[f[n-k+1,k],{n,10},{k,n,1,-1}]//Flatten (*sequence *)
Table[Join[{1},Table[0,4n-1]],{n,10}]//Flatten (* Harvey P. Dale, Dec 21 2016 *)

{a(n) = issquare(2*n + 1)}; /* Michael Somos, Apr 13 2005 */

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^8 + A)^2 / eta(x^4 + A), n))};

PARI
```
A(i,j)=i==j
```

n-th 1 is followed by 4*n-1 0's. In the sequence with flattened indices, the 1's are at positions listed in A046092.
G.f.: 1/(1 - x*y). E.g.f.: exp(x*y).
Considered as a linear sequence, expansion of q^(-1/2)*eta(q^8)^2/eta(q^4) in powers of q. If A(x) is the g.f., then B(a) = (q*A(a^2))^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*w - v^3 - 4*v*w^2. Also, given g.f. A(x), then B(q) = q*A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u2^2*u6 - u1*u6^3 - u3^3*u2. - Michael Somos, Apr 13 2005
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 if p>2. - Michael Somos, Jun 06 2005
a(n) = floor(sqrt(2*n+1)) - floor(sqrt(2*n)). - Ridouane Oudra, Oct 09 2020

A202673 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A115263 based on (1,2,3,4,...); by antidiagonals.

Original entry on oeis.org

1, -1, 1, -6, 1, 1, -12, 20, -1, 1, -18, 75, -50, 1, 1, -24, 166, -328, 105, -1, 1, -30, 293, -1050, 1134, -196, 1, 1, -36, 456, -2432, 5140, -3312, 336, -1, 1, -42, 655, -4690, 15471, -20814, 8514, -540, 1, 1, -48, 890, -8040, 36771, -80584

Offset: 1

Clark Kimberling, Dec 22 2011

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix of A115262 (when A115262 is formatted as a square matrix). The zeros of p(n) are positive, and they interlace the zeros of p(n+1).

			The 1st principal submatrix (ps) of A115263 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{1,2},{2,5}}, with p(2)=1-6x+x^2 and zero-set {0.171..., 5.828...}.
...
The 3rd ps is {{1,2,3},{2,5,8},{3,8,14}}, with p(3)=1-12x+20x^2-x^3 and zero-set {0.099..., 0.516..., 19.383...}.
...
Top of the array:
1....-1
1....-6.....1
1...-12....20.....-1
1...-18....75....-50....1
1...-24...166...-328..105..-1

S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.
A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Cf. A115262, A202671 (based on n^2), A202605 (based on Fibonacci numbers)

U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k, {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]

A271916 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = f(m,n) if m <= n or f(n,m) if n < m, where f(m,n) = m(m-1)(3*n-m-1)/6.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 5, 3, 0, 0, 4, 8, 8, 4, 0, 0, 5, 11, 14, 11, 5, 0, 0, 6, 14, 20, 20, 14, 6, 0, 0, 7, 17, 26, 30, 26, 17, 7, 0, 0, 8, 20, 32, 40, 40, 32, 20, 8, 0, 0, 9, 23, 38, 50, 55, 50, 38, 23, 9, 0, 0, 10, 26, 44, 60, 70, 70, 60, 44, 26, 10, 0

Offset: 1

N. J. A. Sloane, Apr 26 2016

T(m,n) is the number of ways to choose four distinct points from an m X n rectangular grid that form a square aligned with the axes. See A271917 for the count of all subsquares.

			The array begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
0, 2, 5, 8, 11, 14, 17, 20, 23, 26, ...
0, 3, 8, 14, 20, 26, 32, 38, 44, 50, ...
0, 4, 11, 20, 30, 40, 50, 60, 70, 80, ...
0, 5, 14, 26, 40, 55, 70, 85, 100, 115, ...
0, 6, 17, 32, 50, 70, 91, 112, 133, 154, ...
0, 7, 20, 38, 60, 85, 112, 140, 168, 196, ...
0, 8, 23, 44, 70, 100, 133, 168, 204, 240, ...
0, 9, 26, 50, 80, 115, 154, 196, 240, 285, ...
...
As a triangle:
0,
0, 0,
0, 1, 0,
0, 2, 2, 0,
0, 3, 5, 3, 0,
0, 4, 8, 8, 4, 0,
0, 5, 11, 14, 11, 5, 0,
0, 6, 14, 20, 20, 14, 6, 0,
0, 7, 17, 26, 30, 26, 17, 7, 0,
0, 8, 20, 32, 40, 40, 32, 20, 8, 0,
...

See A115262 for another version.
Main diagonal is A000330 (shifted).
Cf. A227133, A271917.

f1:=(m,n)->(1/6)*m*(m-1)*(3*n-m-1);
f2:=(m,n)->if n>=m then f1(m,n) else f1(n,m) fi;
for m from 1 to 10 do
lprint([seq(f2(m,n),n=1..10)]); od;

A187017 Array where the n-th row is the expansion of (1+2x+...+nx^(n-1)) * (n+(n-1)*x+...+x^(n-1)).

Original entry on oeis.org

1, 2, 5, 2, 3, 8, 14, 8, 3, 4, 11, 20, 30, 20, 11, 4, 5, 14, 26, 40, 55, 40, 26, 14, 5, 6, 17, 32, 50, 70, 91, 70, 50, 32, 17, 6, 7, 20, 38, 60, 85, 112, 140, 112, 85, 60, 38, 20, 7, 8, 23, 44, 70, 100, 133, 168, 204, 168, 133, 100, 70, 44, 23, 8, 9, 26, 50, 80, 115, 154, 196, 240, 285, 240, 196, 154, 115, 80, 50, 26, 9

Offset: 1

Franklin T. Adams-Watters, Mar 02 2011

			The table starts:
1
2 5 2
3 8 14 8 3
4 11 20 30 20 11 4

Cf. A005408(row lengths), A000537(row sums), A115262(a rearrangement), A000330(central diagonal).

al(n)=Vec(sum(k=1,n,k*x^(k-1))*sum(k=1,n,k*x^(n-k)))

cp's OEIS Frontend

A202605 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the Fibonacci self-fusion matrix (A202453).

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A185957 Second accumulation array of the array min{n,k}, by antidiagonals.

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A003982 Table read by rows: 1 if x = y, 0 otherwise, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...

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Formula

A202673 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A115263 based on (1,2,3,4,...); by antidiagonals.

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Mathematica

A271916 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = f(m,n) if m <= n or f(n,m) if n < m, where f(m,n) = m*(m-1)*(3*n-m-1)/6.

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Maple

A187017 Array where the n-th row is the expansion of (1+2*x+...+n*x^(n-1)) * (n+(n-1)*x+...+x^(n-1)).

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PARI

A271916 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = f(m,n) if m <= n or f(n,m) if n < m, where f(m,n) = m(m-1)(3*n-m-1)/6.

A187017 Array where the n-th row is the expansion of (1+2x+...+nx^(n-1)) * (n+(n-1)*x+...+x^(n-1)).