A076756 -id:A076756 - cp's OEIS frontend

A054142 Triangular array binomial(2*n-k, k), k=0..n, n >= 0.

1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 15, 10, 1, 1, 9, 28, 35, 15, 1, 1, 11, 45, 84, 70, 21, 1, 1, 13, 66, 165, 210, 126, 28, 1, 1, 15, 91, 286, 495, 462, 210, 36, 1, 1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1

Offset: 0

Clark Kimberling

Row sums are odd-indexed Fibonacci numbers.
T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k double rises. Mirror image of A085478. - Emeric Deutsch, May 31 2004
Diagonal sums are A052535. - Paul Barry, Jan 21 2005
Matrix inverse is the triangle of Salie numbers A098435. - Paul Barry, Jan 21 2005
Coefficients of Morgan-Voyce polynomial b(n,x); e.g., b(3,x)=x^3+5x^2+6x+1. See A172431 for coefficients of Morgan-Voyce polynomial B(n,x). - Clark Kimberling, Feb 13 2010
T(n,k) is the number of stack polyominoes of perimeter 2n+4 with k+1 columns. - Emanuele Munarini, Apr 07 2011
Roots of signed n-th polynomials are chaotic with respect to the operation (-2, x^2), with cycle lengths A003558(n). Example: starting with a root to x^3 - 5x^2 + 6x - 1 = 0; (2 + 2*cos(2*Pi/N) = 3.24697... = A116415; we obtain the trajectory (3.24697...-> 1.55495...-> 0.198062...; the 3 roots to the polynomial with cycle length 3 matching A003558(3) = 3. The operation (-2, x^2) is the reversal of the well known chaotic operation (x^2 - 2) [Kappraff, Adamson, 2004] starting with seed 2*cos(2*Pi/N). Check: given 2*cos(2*Pi/7) = 1.24697..., we obtain the 3-cycle using (x^2 - 2): (1.24697...-> -0.445041...-> 1.801937...; where the terms in either set are intermediate terms in the other, irrespective of sign. - Gary W. Adamson, Sep 22 2011
A054142 is jointly generated with A172431 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=x*u(n-1,x)+v(n-1,x) and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section of A172431. - Clark Kimberling, Mar 09 2012
Subtriangle of the triangle given by (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 01 2012
The o.g.f. for row n of the array A(n, k) = binomial(2*n-k,k), k >= 0, n >= 0 is G(n,x) = Sum_{k=0..n} T(n, k)*x^k + (-x)^(2*n+1) * c(-x)^(2*n+1) / sqrt(1-4*(-x)), for n >= 0. Here c(x) is the o.g.f. of A000108 (Catalan). For powers of c(x) see the W. Lang link in A115139. For the alternating sign case replace x by -x. - Wolfdieter Lang, Sep 12 2016
Multiplying the n-th diagonal by A001147(n) generates A001497. - Tom Copeland, Oct 04 2016

			Triangle begins:
  1;
  1,  1;
  1,  3,  1;
  1,  5,  6,   1;
  1,  7, 15,  10,   1;
  1,  9, 28,  35,  15,   1;
  1, 11, 45,  84,  70,  21,   1;
  1, 13, 66, 165, 210, 126,  28,  1;
  1, 15, 91, 286, 495, 462, 210, 36, 1; ...
...
(0, 1, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, ...) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 3,  1;
  0, 1, 5,  6,  1;
  0, 1, 7, 15, 10,  1;
  0, 1, 9, 28, 35, 15, 1. _Philippe Deléham_, Apr 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..495
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.
Molly Fenn and Eric Sommers, A transitivity result for ad-nilpotent ideals in type A, arXiv:2101.04091 [math.RT], 2021.
J. L. Jacobsen, and J. Salas, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models IV. Chromatic polynomial with cyclic boundary conditions, J. Stat. Phys. 122 (2006) 705-760, arXiv:cond-mat/0407444 See Eq. 2.27. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014
Jay Kappraff and Gary W. Adamson, Polygons and Chaos, 5th Interdispl Symm. Congress and Exh. Jul 8-14, Sydney, 2001 - [with commercial pop-ups].
Jay Kappraff and Gary W. Adamson, Polygons and Chaos, Journal of Dynamical Systems and Geometric Theories, Vol. 2 pp. 79-94, (Nov 2004).

These are the even-indexed rows of A011973, the odd-indexed rows form A053123.
Cf. A000108, A003558, A027989, A052535, A054142, A076756, A084938, A085478, A098435, A115139, A172431, A172991, A188648.
Cf. A001147, A001497.

Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n-k,k) ))); # G. C. Greubel, Aug 01 2019

[Binomial(2*n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

T:=(n,k)->binomial(2*n-k,k): seq(seq(T(n,k), k=0..n), n=0..11);

Flatten[Table[Binomial[2n - k, k], {n, 0, 11}, {k, 0, n}]] (* Emanuele Munarini, Apr 07 2011 *)

create_list(binomial(2*n-k,k),n,0,10,k,0,n); /* Emanuele Munarini, Apr 07 2011 */

T(n,k)=if(n<0,0,polcoeff(charpoly(matrix(n,n,i,j,-min(i,j))),k))

[[binomial(2*n-k,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

G.f.: (1-t*z)/((1-t*z)^2-z). - Emeric Deutsch, May 31 2004
Column k has g.f.: (Sum_{j=0..k+1} binomial(k+1, 2j)*x^j)*x^k/(1-x)^(k+1). - Paul Barry, Jun 22 2005
Recurrence: T(n+2,k+2) = T(n+1,k+2) + 2*T(n+1,k+1) - T(n,k). - Emanuele Munarini, Apr 07 2011
T(n, k) = binomial(2*n-k, k) = A085478(n, n-k), for n >= 0, k = 0..n. - Wolfdieter Lang, Mar 25 2020

A123970 Triangle read by rows: T(0,0)=1; T(n,k) is the coefficient of x^(n-k) in the monic characteristic polynomial of the n X n matrix (min(i,j)) (i,j=1,2,...,n) (0 <= k <= n, n >= 1).

Original entry on oeis.org

1, 1, -1, 1, -3, 1, 1, -6, 5, -1, 1, -10, 15, -7, 1, 1, -15, 35, -28, 9, -1, 1, -21, 70, -84, 45, -11, 1, 1, -28, 126, -210, 165, -66, 13, -1, 1, -36, 210, -462, 495, -286, 91, -15, 1, 1, -45, 330, -924, 1287, -1001, 455, -120, 17, -1, 1, -55, 495, -1716, 3003, -3003, 1820, -680, 153, -19, 1, 1, -66, 715, -3003, 6435, -8008

Offset: 0

Gary W. Adamson and Roger L. Bagula, Oct 29 2006

This sequence is the same as A129818 up to sign. - T. D. Noe, Sep 30 2011

Riordan array (1/(1-x), -x/(1-x)^2). - Philippe Deléham, Feb 18 2012

			Triangular sequence (gives the odd Tutte-Beraha constants as roots!) begins:
  1;
  1,  -1;
  1,  -3,   1;
  1,  -6,   5,   -1;
  1, -10,  15,   -7,    1;
  1, -15,  35,  -28,    9,    -1;
  1, -21,  70,  -84,   45,   -11,   1;
  1, -28, 126, -210,  165,   -66,  13,   -1;
  1, -36, 210, -462,  495,  -286,  91,  -15,  1;
  1, -45, 330, -924, 1287, -1001, 455, -120, 17, -1;
  ...

S. Beraha, Infinite non-trivial families of maps and chromials, Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1975.
Steven R. Finch, Mathematical Constants (Encyclopedia of Mathematics and its Applications), chapter 5.25.
W. T. Tutte, "More about Chromatic Polynomials and the Golden Ratio." In Combinatorial Structures and their Applications: Proc. Calgary Internat. Conf., Calgary, Alberta, 1969. New York: Gordon and Breach, p. 439, 1969.

Paul Fendley and Vyacheslav Krushkal, Tutte chromatic identities and the Temperley-Lieb algebra, arXiv:0711.0016 [math.CO], 2007-2008.
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174. See page 161.
Eric Weisstein's World of Mathematics, Beraha Constants

Cf. A085478, A076756.
Cf. A109954, A129818, A143858, A165253. - R. J. Mathar, Jan 10 2011
Modulo signs, inverse matrix to A039599.

/* As triangle */ [[(-1)^k*Binomial(n + k, 2*k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jan 04 2019

with(linalg): m:=(i,j)->min(i,j): M:=n->matrix(n,n,m): T:=(n,k)->coeff(charpoly(M(n),x),x,n-k): 1; for n from 1 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

An[d_] := MatrixPower[Table[Min[n, m], {n, 1, d}, {m, 1, d}], -1]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]

f(n,x) = (2x-1)f(n-1,x)-x^2*f(n-2,x), where f(n,x) is the characteristic polynomial of the n X n matrix from the definition and f(0,x)=1. See formula in Fendley and Krushkal. - Jonathan Vos Post, Nov 04 2007
T(n,k) = (-1)^k * A085478(n,k) = (-1)^n * A129818(n,k). - Philippe Deléham, Feb 06 2012
T(n,k) = 2*T(n-1,k) - T(n-1,k-1) - T(n-2,k), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 29 2013

Edited by N. J. A. Sloane, Nov 29 2006

A202672 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A087062 based on (1,1,1,1,...); by antidiagonals.

Original entry on oeis.org

1, -1, 1, -3, 1, 1, -5, 6, -1, 1, -7, 15, -10, 1, 1, -9, 28, -35, 15, -1, 1, -11, 45, -84, 70, -21, 1, 1, -13, 66, -165, 210, -126, 28, -1, 1, -15, 91, -286, 495, -462, 210, -36, 1, 1, -17, 120, -455, 1001, -1287, 924, -330, 45, -1, 1, -19, 153

Offset: 1

Clark Kimberling, Dec 22 2011

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix of A087062. The zeros of p(n) are positive, and they interlace the zeros of p(n+1).

Closely related to A076756; however, for example, successive rows of A076756 are (1,-3,1), (-1,5,-6,1), compared to rows (1,-3,1), (1,-5,6,-1) of A202672.

			The 1st principal submatrix (ps) of A087062 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 {0.381..., 2.618...}.
...
The 3rd ps is {{1,1,1},{1,2,2},{1,2,3}}, with p(3)=1-5x+6x^2-x^3 and zero-set {0.283..., 0.426..., 8.290...}.
...
Top of the array:
1...-1
1...-3....1
1...-5....6....-1
1...-7...15...-10....1
1...-9...28...-35...15...-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. A087062, A202673 (based on n), A202671 (based on n^2), A202605 (based on Fibonacci numbers), A076756.

U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[1, {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}]]
Table[(F[k] /. x -> -2), {k, 1, 30}] (* A007583 *)
Table[(F[k] /. x -> 2), {k, 1, 30}]  (* A087168 *)

A203469 a(n) = v(n)/A000178(n), v = A093883 and A000178 = (superfactorials).

Original entry on oeis.org

1, 3, 30, 1050, 132300, 61122600, 104886381600, 674943865596000, 16407885372638760000, 1515727634953623371280000, 534621388490302221024396480000, 722849817707190846398223943885440000, 3759035907022704558524683975387453632000000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..55

Cf. A000178, A006963, A076756, A093883, A296590.

[(&*[Binomial(2*n-k,k): k in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 29 2023

(* First program *)
f[j_]:= j; z = 16;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,z}]           (* A093883 *)
Table[v[n+1]/v[n], {n,z-1}]  (* A006963 *)
Table[v[n]/d[n], {n,20}]     (* A203469 *)
(* Second program *)
Table[Product[Binomial[2*n-j,j], {j,n}], {n,20}] (* G. C. Greubel, Aug 29 2023 *)

[product(binomial(2*n-j,j) for j in range(n)) for n in range(1, 20)] # G. C. Greubel, Aug 29 2023

a(n) = Product_{i=1..n} binomial(2n-i,i).  - Enrique Pérez Herrero, Feb 20 2013
From G. C. Greubel, Aug 29 2023: (Start)
a(n) = (2^n/sqrt(Pi))^n*BarnesG(n+3/2)/(BarnesG(n+2)*BarnesG(3/2)).
a(n) = (n!/2^(n-1))*Product_{j=1..n-1} Catalan(j). (End)
a(n) ~ A^(3/2) * exp(n/2 - 1/8) * 2^(n^2 - 7/24) / (Pi^(n/2 + 1/2) * n^(n/2 + 3/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 26 2023

A203989 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {max(i,j)} (A051125).

Original entry on oeis.org

1, -1, -2, -3, 1, 3, 11, 6, -1, -4, -23, -35, -10, 1, 5, 39, 98, 85, 15, -1, -6, -59, -207, -308, -175, -21, 1, 7, 83, 374, 795, 798, 322, 28, -1, -8, -111, -611, -1694, -2475, -1806, -546, -36, 1, 9, 143, 930, 3185, 6149, 6633, 3696, 870, 45

Offset: 1

Clark Kimberling, Jan 09 2012

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are real, and they interlace the zeros of p(n+1). See A202605 for a guide to related sequences.

The characteristic polynomial seems be the recurrence relation given by p(n,x) = -x * p(n-1,x) + n * (-1)^(n-1) * sum_{i=0..n-1} x^i * binomial(2n-i-2,i). - Enrique Pérez Herrero, Jan 29 2013

			Top of the array:
1... -1
-2... -3.... 1
3.... 11... 6... -1
-4... -23.. -35.. -10...1
5.... 39... 98... 85...15.. -1

(For references regarding interlacing roots, see A202605.)

Cf. A051125, A202605.

Cf. A181983, A076756.

f[i_, j_] := Max[i, j];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6th principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]  (* A051125 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]              (* A203989 *)
TableForm[Table[c[n], {n, 1, 10}]]

A122073 Triangular array from Steinbach matrices plus their squares as characteristic polynomials: M[i,j]=A[i,j]+A[i,j]^2: A[i,j]^2=Min[i,j]; CharacteristicPolynomial[M[i,j]];.

Original entry on oeis.org

1, 2, -1, 0, -4, 1, 2, -9, 8, -1, -2, -3, 19, -12, 1, -4, -6, 47, -55, 18, -1, 2, 15, 0, -88, 93, -24, 1, 2, 23, -7, -190, 324, -182, 32, -1, 0, -12, -63, 62, 332, -554, 274, -40, 1, 2, -9, -108, 133, 678, -1642, 1346, -450, 50, -1, -2, -11, 55, 276, -463, -1129, 2832, -2128, 630, -60, 1, -4, -30, 71, 543, -1044, -2204, 7761

Offset: 1

Gary W. Adamson and Roger L. Bagula, Oct 16 2006

Based on the idea that the Steinbach matrices form a "golden Field". Matrices are: {{2, 2}, {2, 2}}, {{2, 2, 2}, {2, 3, 2}, {2, 2, 3}}, {{2, 2, 2, 2}, {2, 3, 3, 2}, {2, 3, 3, 3}, {2, 2, 3, 4}}, {{2, 2, 2, 2, 2}, {2, 3, 3, 3, 2}, {2, 3, 4, 3, 3}, {2, 3, 3, 4, 4}, {2, 2, 3, 4, 5}}, {{2, 2, 2, 2, 2, 2}, {2,3, 3, 3, 3, 2}, {2, 3, 4, 4, 3, 3}, {2, 3, 4, 4, 4, 4}, {2, 3, 3, 4, 5, 5}, {2, 2, 3, 4, 5, 6}}

			{1},
{2, -1},
{0, -4, 1},
{2, -9, 8, -1},
{-2, -3, 19, -12, 1},
{-4, -6,47, -55, 18, -1}
{2, 15, 0, -88, 93, -24, 1},
{2, 23, -7, -190, 324, -182, 32, -1},
{0, -12, -63, 62, 332, -554, 274, -40, 1}

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Cf. A038223, A038223, A076756, A054142.

An[d_] := Table[Min[n, m] + If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d,1, 20}]]; Flatten[%]

d-th level M(i,j)->An[d]; T(n,m)=CoefficientList[CharacteristicPolynomial[An[d], x], x]

cp's OEIS Frontend

A054142 Triangular array binomial(2*n-k, k), k=0..n, n >= 0.

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A123970 Triangle read by rows: T(0,0)=1; T(n,k) is the coefficient of x^(n-k) in the monic characteristic polynomial of the n X n matrix (min(i,j)) (i,j=1,2,...,n) (0 <= k <= n, n >= 1).

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A202672 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A087062 based on (1,1,1,1,...); by antidiagonals.

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A203469 a(n) = v(n)/A000178(n), v = A093883 and A000178 = (superfactorials).

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A203989 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {max(i,j)} (A051125).

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A122073 Triangular array from Steinbach matrices plus their squares as characteristic polynomials: M[i,j]=A[i,j]+A[i,j]^2: A[i,j]^2=Min[i,j]; CharacteristicPolynomial[M[i,j]];.

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