A115255 -id:A115255 - cp's OEIS frontend

A203005 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A115255 (in square format); by antidiagonals.

1, -1, 1, -6, 1, 1, -15, 47, -1, 1, -40, 270, -488, 1, 1, -165, 1738, -5866, 5829, -1, 1, -1074, 15695, -80060, 156495, -74674, 1, 1, -9039, 181581, -1360515, 4552003, -5997165, 997295, -1, 1, -86700, 2566036, -28081556

Offset: 1

Clark Kimberling, Dec 27 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 they interlace the zeros of p(n+1).

			Top of the array:
1...-1
1...-6....1
1...-15...47....-1
1...-40...270...-488...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. A115255, A202605.

f[k_] := Binomial[2 k - 2, k - 1];
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}]]

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

Original entry on oeis.org

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}]]

A115257 Partial sums of binomial(2n,n)^2.

Original entry on oeis.org

1, 5, 41, 441, 5341, 68845, 922621, 12701245, 178338145, 2542242545, 36677022081, 534311328705, 7846771001041, 116019251361041, 1725360846921041, 25786805857871441, 387084441100423541, 5832802431123111941

Offset: 0

Paul Barry, Jan 18 2006

Central coefficients of number triangle A115255.
p divides all a(n) from a((p-1)/2) to a(p-1) for Gaussian primes p=7,23,31,79,167,431,479,983, ... of the form 4n+3, A002145(n) and for primes of the form 8n+7, A007522(n). - Alexander Adamchuk, Jul 05 2006
Conjecture: For any positive integer n, the polynomials Sum_{k=0}^n binomial(2k,k)^2*x^k and Sum_{k=0}^n binomial(2k,k)^2*x^k/(k+1) are irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 23 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
DLMF Digital Library of Mathematical Functions, Elliptic Integrals, NIST, 2016.

Cf. A000984, A002145, A007522, A115255, A228002.

series( 2*EllipticK(4*x^(1/2))/(Pi*(1-x)) ,x=0,20); # Mark van Hoeij, Apr 06 2013

Table[Sum[((2k)!/(k!)^2)^2,{k,0,n}], {n,0,40}] (* Alexander Adamchuk, Jul 05 2006 *)
Accumulate[(Binomial[2#,#])^2&/@Range[0,20]]  (* Harvey P. Dale, Mar 04 2011 *)

makelist(sum(binomial(2*k,k)^2,k,0,n),n,0,12); /* Emanuele Munarini, Oct 28 2016 */

a(n) = sum(k=0, n, binomial(2*k, k)^2); \\ Michel Marcus, Oct 30 2016

a(n) = Sum_{k=0..n} C(2k, k)^2. a(n) = A115255(2n, n).
a(n) = C(2n,n)^2 + C(2n-2,n-1)^2 + ... + C(2k,k)^2 + ... + C(2,1)^2 + C(0,0)^2, where C(2k,k) = (2k)!/(k!)^2 are the central binomial coefficients A000984(k). - Alexander Adamchuk, Jul 05 2006
a(n) = Sum_{k=0..n} ((2k)!/(k!)^2)^2. a(n) = Sum_{k=0..n} A000984[k]^2. - Alexander Adamchuk, Jul 05 2006
Recurrence: n^2*a(n) = (17*n^2-16*n+4)*a(n-1) - 4*(2*n-1)^2*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 16^(n+1)/(15*Pi*n). - Vaclav Kotesovec, Oct 19 2012
From Emanuele Munarini, Oct 28 2016: (Start)
Let K(x) be the complete elliptic integral of the first kind as defined in [DLMF, 19.2.4] for phi = Pi/2.
a(n) = (2/Pi)*K(16)-((16^(n+1)*Gamma(n+3/2)^2)/(Pi*Gamma(n+2)^2))*hypergeometric (1,n+3/2,n+3/2;n+2,n+2;16).
G.f.: A(t) = (2/Pi)*(K(16*t)/(1-t)).
Diff. eq. satisfied by the g.f. t*(1-17*t+16*t^2)*A''(t)+(1-35*t+64*t^2)*A'(t)-(5-36*t)*A(t)=0. (End)

A115256 Diagonal sums of correlation triangle of central binomial coefficients.

Original entry on oeis.org

1, 2, 8, 25, 90, 312, 1145, 4186, 15640, 58681, 222298, 845848, 3235385, 12418650, 47827992, 184688185, 714884186, 2772776984, 10774163001, 41932100698, 163430680600, 637793652281, 2491918144602, 9746480252952, 38157725306425

Offset: 0

Paul Barry, Jan 18 2006

Diagonal sums of number triangle A115255.

G. C. Greubel, Table of n, a(n) for n = 0..1000

CoefficientList[Series[1/((Sqrt[1-4x])(Sqrt[1-4x^2])(1-x^3)), {x,0,30}], x] (* Harvey P. Dale, Feb 15 2012 *)

my(x='x+O('x^50)); Vec(1/(sqrt(1-4*x)*sqrt(1-4*x^2)*(1-x^3))) \\ G. C. Greubel, Mar 18 2017

G.f.: 1/(sqrt(1-4*x)*sqrt(1-4*x^2)*(1-x^3)).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} [j<=k]*C(2*k-2*j, k-j)*[j<=n-2*k]*C(2*n-4*k-2*j, n-2*k-j).
a(n) ~ sqrt(3) * 2^(2*n+7) / (189 * sqrt(Pi*n)). - Vaclav Kotesovec, Mar 02 2014
Conjecture: n*a(n) + 2*(-2*n+1)*a(n-1) + 4*(-n+1)*a(n-2) + 3*(5*n-8)*a(n-3) + 2*(2*n-1)*a(n-4) + 4*(n-1)*a(n-5) + 8*(-2*n+3)*a(n-6) = 0. - R. J. Mathar, Jun 22 2016

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A203005 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A115255 (in square format); by antidiagonals.

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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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A115257 Partial sums of binomial(2n,n)^2.

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A115256 Diagonal sums of correlation triangle of central binomial coefficients.

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