A054142 -id:A054142 - cp's OEIS frontend

A116425 Decimal expansion of 2 + 2cos(2Pi/7).

3, 2, 4, 6, 9, 7, 9, 6, 0, 3, 7, 1, 7, 4, 6, 7, 0, 6, 1, 0, 5, 0, 0, 0, 9, 7, 6, 8, 0, 0, 8, 4, 7, 9, 6, 2, 1, 2, 6, 4, 5, 4, 9, 4, 6, 1, 7, 9, 2, 8, 0, 4, 2, 1, 0, 7, 3, 1, 0, 9, 8, 8, 7, 8, 1, 9, 3, 7, 0, 7, 3, 0, 4, 9, 1, 2, 9, 7, 4, 5, 6, 9, 1, 5, 1, 8, 8, 5, 0, 1, 4, 6, 5, 3, 1, 7, 0, 7, 4, 3, 3, 3, 4, 1, 1

Offset: 1

Eric W. Weisstein, Feb 15 2006

A root of the equation x^3 - 5*x^2 + 6*x - 1 = 0. - Arkadiusz Wesolowski, Jan 13 2016

The other two roots of this minimal polynomial of the present algebraic number (rho(7))^2, with rho(7) = 2*cos(Pi/7) = A160389 are (2*cos(3*Pi/7))^2 = (A255241)^2 and (2*cos(5*Pi/7))^2 = (-A255249)^2. - Wolfdieter Lang, Mar 30 2020

			3.246979603717467061...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.25 Tutte-Beraha Constants, p. 417.

Jesús Salas and Alan D. Sokal, Transfer matrices and partition functions zeros for antiferromagnetic Potts models, arXiv:cond-mat/0004330 [cond-mat.stat-mech], 2000-2001, p. 64.
Eric Weisstein's World of Mathematics, Logistic Map
Eric Weisstein's World of Mathematics, Silver Constant
Index entries for algebraic numbers, degree 3

Cf. A231187, A160389.
Cf. A003558, A054142, A255241, A255249.
2 + 2*cos(2*Pi/n): A104457 (n = 5), A332438 (n = 9), A296184 (n = 10), A019973 (n = 12).

First@ RealDigits[N[2 + 2 Cos[2 Pi/7], 120]] (* Michael De Vlieger, Jan 13 2016 *)

2 + 2*cos(2*Pi/7) \\ Michel Marcus, Jan 13 2016

Equals (2*cos(Pi/7))^2 = (A160389)^2.
Equals 2 + i^(4/7) - i^(10/7). - Peter Luschny, Apr 04 2020
Let  c = 2 + 2*cos(2*Pi/7). The linear fractional transformation z -> c - c/z has order 7, that is, z = c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/z)))))). - Peter Bala, May 09 2024

A052535 Expansion of (1-x)(1+x)/(1-x-2x^2+x^4).

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 50, 95, 181, 345, 657, 1252, 2385, 4544, 8657, 16493, 31422, 59864, 114051, 217286, 413966, 788674, 1502555, 2862617, 5453761, 10390321, 19795288, 37713313, 71850128, 136886433, 260791401, 496850954, 946583628

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n) = number of compositions of n with parts in {2,1,3,5,7,9,...}. The generating function follows easily from Theorem 1.1 of the Hoggatt et al. reference. Example: a(4)= 7 because we have 22, 31, 13, 211, 121, 112, and 1111. - Emeric Deutsch, Aug 17 2016.
Diagonal sums of A054142. - Paul Barry, Jan 21 2005
Equals INVERT transform of (1, 1, 1, 0, 1, 0, 1, 0, 1, ...). - Gary W. Adamson, Apr 27 2009
Number of tilings of a 4 X 2n rectangle by 4 X 1 tetrominoes. - M. Poyraz Torcuk, Dec 10 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024. See p. 18.
V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 465
Todd Mullen, On Variants of Diffusion, Dalhousie University (Halifax, NS Canada, 2020).
Todd Mullen, Richard Nowakowski, and Danielle Cox, Counting Path Configurations in Parallel Diffusion, arXiv:2010.04750 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1).

Cf. A275446.
Bisection of A003269 (odd part),
Cf. A236582, A251074, A236580.

a:=[1,1,2,4];; for n in [5..40] do a[n]:=a[n-1]+2*a[n-2]-a[n-4]; od; a; # G. C. Greubel, May 09 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2)/( 1-x-2*x^2+x^4) )); // G. C. Greubel, May 09 2019

spec := [S,{S=Sequence(Prod(Z,Union(Z,Sequence(Prod(Z,Z)))))},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

CoefficientList[Series[(1-x^2)/(1-x-2x^2+x^4), {x, 0, 40}], x] (* or *)
Table[Length@ Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {_, a_, _} /; And[EvenQ@ a, a != 2]]], 1], {n, 0, 40}]  (* Michael De Vlieger, Aug 17 2016 *)
LinearRecurrence[{1,2,0,-1},{1,1,2,4},40] (* Harvey P. Dale, Apr 12 2018 *)

my(x='x+O('x^40)); Vec((1-x^2)/(1-x-2*x^2+x^4)) \\ G. C. Greubel, May 09 2019

((1-x^2)/(1-x-2*x^2+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

G.f.: (1 - x^2)/(1 - x - 2*x^2 + x^4).
a(n) = a(n-1) + 2*a(n-2) - a(n-4), with a(0)=1, a(1)=1, a(2)=2, a(3)=4.
a(n) = Sum_{alpha = RootOf(1-x-2*x^2+x^4)} (1/283)*(27 + 112*alpha + 9*alpha^2 -48*alpha^3)*alpha^(-n-1).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k, k). - Paul Barry, Jan 21 2005
a(n) = A158943(n) -A158943(n-2). - R. J. Mathar, Jan 13 2023

More terms from James Sellers, Jun 05 2000

A169803 Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 3, 0, 0, 1, 5, 6, 1, 0, 0, 1, 6, 10, 4, 0, 0, 0, 1, 7, 15, 10, 1, 0, 0, 0, 1, 8, 21, 20, 5, 0, 0, 0, 0, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0, 1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0, 1, 12, 55, 120, 126, 56, 7, 0, 0, 0, 0, 0, 0

Offset: 0

Nadia Heninger and N. J. A. Sloane, May 21 2010

T(n,k) = 0 if k <0 or k > n+1-k.
T(n,k) is the number of binary vectors of length n and weight k containing no pair of adjacent 1's.
Take Pascal's triangle A007318 and push the k-th column downwards by 2k-1 places (k>=1).
Row sums are A000045.
From Emanuele Munarini, May 24 2011: (Start)
Diagonal sums are A000930(n+1).
A sparse subset (or scattered subset) of {1,2,...,n} is a subset never containing two consecutive elements. T(n,k) is the number of sparse subsets of {1,2,...,n} having size k. For instance, for n=4 and k=2 we have the 3 sparse 2-subsets of {1,2,3,4}: 13, 14, 24. (End)
As a triangle, row 2*n-1 consists of the coefficients of Morgan-Voyce polynomial B(n,x), A172431, and row 2*n to the coefficients of Morgan-Voyce polynomial b(n,x), A054142.
Aside from signs and index shift, the coefficients of the characteristic polynomial of the Coxeter adjacency matrix for the Coxeter group A_n related to the Chebyshev polynomial of the second kind (cf. Damianou link p. 19). - Tom Copeland, Oct 11 2014
Antidiagonals of the Pascal matrix A007318 read bottom to top, omitting the first antidiagonal. These are also the antidiagonals (omitting the first antidiagonal) read from top to bottom of the numerical coefficients of the Maurer-Cartan form matrix of the Leibniz group L^(n)(1,1) presented on p. 9 of the Olver paper, which is generated as exp[c. * M] with (c.)^n = c_n and M the Lie infinitesimal generator A218272. Reverse is embedded in A102426. - Tom Copeland, Jul 02 2018

			Triangle begins:
  [1]
  [1, 1]
  [1, 2, 0]
  [1, 3, 1, 0]
  [1, 4, 3, 0, 0]
  [1, 5, 6, 1, 0, 0]
  [1, 6, 10, 4, 0, 0, 0]
  [1, 7, 15, 10, 1, 0, 0, 0]
  [1, 8, 21, 20, 5, 0, 0, 0, 0]
  [1, 9, 28, 35, 15, 1, 0, 0, 0, 0]
  [1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0]
  [1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0]
  [1, 12, 55, 120, 126, 56, 7, 0, 0, 0, 0, 0, 0]
  [1, 13, 66, 165, 210, 126, 28, 1, 0, 0, 0, 0, 0, 0]
  [1, 14, 78, 220, 330, 252, 84, 8, 0, 0, 0, 0, 0, 0, 0]
  [1, 15, 91, 286, 495, 462, 210, 36, 1, 0, 0, 0, 0, 0, 0, 0]
  [1, 16, 105, 364, 715, 792, 462, 120, 9, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
  ...

Indranil Ghosh, Rows 0..125, flattened
Pantelis A. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
Emanuele Munarini and Norma Zagaglia Salvi, Scattered Subsets, Discrete Mathematics 267 (2003), 213-228.
Emanuele. Munarini and Norma Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
Emanuele Munarini, A combinatorial interpretation of the Chebyshev polynomials, SIAM Journal on Discrete Mathematics, Volume 20, Issue 3 (2006), 649-655.
Peter J. Olver, The canonical contact form.
James J. Y. Zhao, Infinite log-concavity and higher order Turán inequality for Speyer's g-polynomial of uniform matroids, arXiv:2409.08085 [math.CO], 2024. See p. 11.

Cf. A000045, A000930, A007318, A011973 (another version), A218272.
All of A011973, A092865, A098925, A102426, A169803 describe essentially the same triangle in different ways. - N. J. A. Sloane, May 29 2011
A172431 and A054142 describe the odd and even lines of the triangle.

T[n_,k_]:= Binomial[n+1-k,k]; Table[T[n,k],{n,0,12},{k,0,n}]//Flatten (* Stefano Spezia, Sep 16 2024 *)

create_list(binomial(n-k+1,k),n,0,20,k,0,n); /* Emanuele Munarini, May 24 2011 */

T(n,k)=binomial(n+1-k,k) \\ Charles R Greathouse IV, Oct 24 2012

A121314 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

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, 0, 1, 11, 45, 84, 70, 21, 1, 0, 1, 13, 66, 165, 210, 126, 28, 1, 0, 1, 15, 91, 286, 495, 462, 210, 36, 1

Offset: 0

Philippe Deléham, Aug 25 2006

A054142 with first diagonal 1, 0, 0, 0, 0, 0, 0, 0, ...

Mirror image of triangle in A165253.

			Triangle 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;
  0,  1, 11, 45, 84, 70, 21,  1;

Alois P. Heinz, Rows n = 0..140, flattened
F. Yano and H. Yoshida, Some set partition statistics in non-crossing partitions and generating functions, Discr. Math., 307 (2007), 3147-3160.

Cf. A054142, A165253.

T(0,0)=1; T(n,0)=0 for n > 0; T(n+1,k+1) = binomial(2*n-k,k)for n >= 0 and k >= 0.
Sum_{k=0..n} T(n,k)*x^k = A001519(n), A047849(n), A165310(n), A165311(n), A165312(n), A165314(n), A165322(n), A165323(n), A165324(n) for x = 1,2,3,4,5,6,7,8,9 respectively.
Sum_{k=0..n} 2^k*T(n,k) = (4^n+2)/3.
Sum_{k=0..n} 2^(n-k)*T(n,k) = A001835(n).
Sum_{k=0..n} 3^k*4^(n-k)*T(n,k) = A054879(n). - Philippe Deléham, Aug 26 2006
Sum_{k=0..n} T(n,k)*(-1)^k*2^(3n-2k) = A143126(n). - Philippe Deléham, Oct 31 2008
Sum_{k=0..n} T(n,k)*(-1)^k*3^(n-k) = A138340(n)/4^n. - Philippe Deléham, Nov 01 2008
G.f.: (1-(y+1)*x)/(1-(2y+1)*x+y^2*x^2). - Philippe Deléham, Nov 01 2011
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0. - Philippe Deléham, Feb 19 2012

A172431 Even row Pascal-square read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 10, 4, 1, 8, 21, 20, 5, 1, 10, 36, 56, 35, 6, 1, 12, 55, 120, 126, 56, 7, 1, 14, 78, 220, 330, 252, 84, 8, 1, 16, 105, 364, 715, 792, 462, 120, 9, 1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 10

Offset: 1

Mark Dols, Feb 02 2010

Apart from signs identical to A053123. Mirror of A078812.
As a triangle, row n consists of the coefficients of Morgan-Voyce polynomial B(n,x); e.g., B(3,x)=x^3+6x^2+10x+4. As a triangle, rows 0 to 4 are as follows: 1 1...2 1...4...3 1...6...10...4 1...8...21...20...5 See A054142 for coefficients of Morgan-Voyce polynomial b(n,x).
Scaled version of A119900. - Philippe Deléham, Feb 24 2012
A172431 is jointly generated with A054142 as an array of coefficients of polynomials v(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. - Clark Kimberling, Mar 09 2012
Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 22 2012

			Array begins:
  1,  2,  3,  4,  5,  6, ...
  1,  4, 10, 20, 35, ...
  1,  6, 21, 56, ...
  1,  8, 36, ...
  1, 10, ...
  1, ...
  ...
Example:
Starting with 1, every entry is twice the one to the left minus the second one to the left, plus the one above.
For n = 9 the a(9) = 10 solution is 2*4 - 1 + 3.
From _Philippe Deléham_, Feb 24 2012: (Start)
Triangle T(n,k) begins:
  1;
  1,   2;
  1,   4,   3;
  1,   6,  10,   4;
  1,   8,  21,  20,   5;
  1,  10,  36,  56,  35,   6;
  1,  12,  55, 120, 126,  56,   7; (End)
From _Philippe Deléham_, Mar 22 2012: (Start)
(1, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, 1/2, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   4,   3,   0;
  1,   6,  10,   4,   0;
  1,   8,  21,  20,   5,   0;
  1,  10,  36,  56,  35,   6,   0;
  1,  12,  55, 120, 126,  56,   7,   0; (End)

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A078812, A053123, A007318, A001906 (antidiagonals sums), A007685.

Cf. also A054142, A082985.

F:=Factorial;; Flat(List([1..15], n-> List([1..n], k-> Sum([0..Int((k-1)/2)], j-> (-1)^j*F(n-j-1)*2^(k-2*j-1)/(F(j)*F(n-k)*F(k-2*j-1)) )))); # G. C. Greubel, Dec 15 2019

F:=Factorial; [ &+[(-1)^j*F(n-j-1)*2^(k-2*j-1)/(F(j)*F(n-k)*F(k-2*j-1)): j in [0..Floor((k-1)/2)]]: k in [1..n], n in [1..15]]; // G. C. Greubel, Dec 15 2019

T := (n, k) -> simplify(GegenbauerC(k, n-k, 1)):
for n from 0 to 10 do seq(T(n,k), k=0..n-1) od; # Peter Luschny, May 10 2016

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A054142 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A172431 *)
(* Clark Kimberling, Mar 09 2012 *)
Table[GegenbauerC[k-1, n-k+1, 1], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Dec 15 2019 *)

T(n,k) = sum(j=0, (k-1)\2, (-1)^j*(n-j-1)!*2^(k-2*j-1)/(j!*(n-k)!*(k-2*j-1)!) );
for(n=1, 10, for(k=1, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 15 2019

[[gegenbauer(k-1, n-k+1, 1) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Dec 15 2019

As a decimal sequence: a(n)= 12*a(n-1)- a(n-2) with a(1)=1. [I interpret this remark as: 1, 12=1,2, 143=1,4,3, 1704=1,6,10,4,... taken from A004191 are decimals on the diagonal. - R. J. Mathar, Sep 08 2013]
As triangle T(n,k): T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2). - Philippe Deléham, Feb 24 2012
As DELTA-triangle T(n,k) with 0<=k<=n: G.f.: (1-y*x)^2/((1-y*x)^2-x). - Philippe Deléham, Mar 22 2012
T(n, k) = GegenbauerC(k, n-k, 1). - Peter Luschny, May 10 2016
As triangle T(n,k): Product_{k=1..n} T(n,k) = Product_{k=0..n-1} binomial(2*k,k) = A007685(n-1)  for n >= 1. - Werner Schulte, Apr 26 2017
As triangle T(n,k) with 1 <= k <= n: T(n,k) = binomial(2*n-k, k-1). - Paul Weisenhorn, Nov 25 2019

A332602 Tridiagonal matrix M read by antidiagonals: main diagonal is 1,2,2,2,2,..., two adjacent diagonals are 1,1,1,1,1,...

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Mar 06 2020, following a suggestion from Gary W. Adamson

From Gary W. Adamson, Mar 11 2020: (Start)
The upper left entry of M^n gives the Catalan numbers A000108. Extracting 2 X 2, 3 X 3, and 4 X 4 submatrices from M; then generating sequences from the upper left entries of M^n, we obtain the following sequences:
1, 1, 2, 5, 13, ... = A001519 and the convergent is 2.61803... = 2 + 2*cos(2*Pi/5) = (2*cos(Pi/5))^2.
1, 1, 2, 5, 14, 42, 131, ... = A080937 and the convergent is 3.24697... = 2 + 2*cos(2*Pi/7) = (2*cos(Pi/7))^2.
1, 1, 2, 5, 14, 42, 132, 429, 1429, ... = A080938 and the convergent is 3.53208... = 2 + 2*cos(2*Pi/9) = (2*cos(Pi/9))^2. (End)
The characteristic polynomial for the N X N main submatrix M_N is Phi(N, x) = S(N, 2-x) - S(N-1, 2-x), with Chebyshev's S polynomial (see A049310) evaluated at 2-x. Proof by determinant expansion, to obtain the recurrence Phi(N, x) - (x-2)*Phi(N-1, x) - Phi(N-2, x), for N >= 2, and Phi(0, x) = 1 and Phi(1, x) = 1 - x, that is Phi(-1, x) = 1. The trace is tr(M_N) = 1 + 2^(N-1) = A000051(N-1), and Det(M_N) = 1. - Wolfdieter Lang, Mar 13 2020
The explicit form of the characteristic polynomial for the N X N main submatrix M_N is Phi(N, x) := Det(M_N - x*1_N) = Sum_{k=0..N} binomial(N+k, 2*k)*(-x)^k = Sum_{k=0..N} A085478(N, k)*(-x)^k, for N >= 0, with Phi(0, x) := 1. Proof from the recurrence given in the preceding comment. - Wolfdieter Lang, Mar 25 2020
For the proofs of the 2 X 2, 3 X 3 and 4 X 4 conjectures, see the comments in the respective A-numbers A001519, A080937 and A080938. - Wolfdieter Lang, Mar 30 2020
Replace the main diagonal 1,2,2,2,... of the matrix M with 1,0,0,0,...; 1,1,1,1,...; 1,3,3,3,...; 1,2,1,2,...; 1,2,3,4,...; 1,0,1,0...; and 1,1,0,0,1,1,0,0,.... Take powers of M and extract the upper left terms, resulting in respectively: A001405, A001006, A033321, A176677, A006789, A090344, and A007902. - Gary W. Adamson, Apr 12 2022
The statement that the upper left entry of M^n is a Catalan number is equivalent to Exercise 41 of R. Stanley, "Catalan Numbers." - Richard Stanley, Feb 28 2023
If the upper left 1 in matrix M is replaced with 3, taking powers of the resulting matrix and extracting the upper left terms apparently results in sequence A001700. - Gary W. Adamson, Apr 03 2023

			The matrix begins:
  1, 1, 0, 0, 0, ...
  1, 2, 1, 0, 0, ...
  0, 1, 2, 1, 0, ...
  0, 0, 1, 2, 1, ...
  0, 0, 0, 1, 2, ...
  ...
The first few antidiagonals are:
  1;
  1, 1;
  0, 2, 0;
  0, 1, 1, 0;
  0, 0, 2, 0, 0;
  0, 0, 1, 1, 0, 0;
  0, 0, 0, 2, 0, 0, 0;
  0, 0, 0, 1, 1, 0, 0, 0;
  0, 0, 0, 0, 2, 0, 0, 0, 0;
  0, 0, 0, 0, 1, 1, 0, 0, 0, 0;
  ...
Characteristic polynomial of the 3 X 3 matrix M_3: Phi(3, x) = 1 - 6*x + 5*x^2 - x^3, from {A085478(3, k)}_{k=0..3} = {1, 6, 5, 1}. - _Wolfdieter Lang_, Mar 25 2020

Richard P. Stanley, "Catalan Numbers", Cambridge University Press, 2015.

Cf. A000108, A000051, A001519, A049310, A080937, A080938, A085478.
Cf. A001006, A001405, A006789, A033321, A176677, A090344, A007902.
Cf. A001333 (permanent of the matrix M).
Cf. A054142, A053123, A011973 (characteristic polynomials of submatrices of M).
Cf. A001700.

A076756 Triangle of coefficients of characteristic polynomial of M_n, the n X n matrix M_(i,j) = min(i,j).

Original entry on oeis.org

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

Benoit Cloitre, Nov 09 2002

The characteristic polynomial of M_n seems to be p(n,x) = (-1)^n * sum_{i=0..n} (-x)^i * binomial(2n-i, i). - Enrique Pérez Herrero, Jan 29 2013

			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;

Alois P. Heinz, Rows n = 0..140, flattened (rows 0..60 from Enrique Pérez Herrero)
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. - _N. J. A. Sloane_, Mar 14 2014

Absolute values are A054142.

Cf. A203989.

T:=(n, k)-> binomial(2*n-k, k)*(-1)^(n+k):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Feb 01 2013

T[n_, k_] := Binomial[2*n - k, k]*(-1)^(n + k); Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jun 12 2015, after Alois P. Heinz *)

Offset corrected by Alois P. Heinz, Feb 01 2013

A188648 Binomial sums a(n) = Sum_{k=0..n} (binomial(2n-k,k))^2.

Original entry on oeis.org

1, 2, 11, 63, 376, 2317, 14545, 92512, 594169, 3844787, 25027296, 163701327, 1075049011, 7083830648, 46812088751, 310118453573, 2058919125662, 13695571200353, 91254952276859, 608960974528058, 4069232436916151

Offset: 0

Emanuele Munarini, Apr 07 2011

Central coefficients of A172991.

Bisection of A051286 (Whitney number of level n of the lattice of the ideals of the fence of order 2n). - Paul D. Hanna, Apr 07 2011

Seiichi Manyama, Table of n, a(n) for n = 0..1198 (terms 0..121 from Vincenzo Librandi)

Sum_{k=0..n} (binomial(2n-k,k))^b: A122367(n) = A001519(n+1) (b=1), this sequence (b=2).

Cf. A054142, A172991, A051286.

Table[Sum[Binomial[2n-k,k]^2,{k,0,n}],{n,0,20}]
Table[DifferenceRoot[Function[{y, m}, {4 (-m + n)^2 (-1 - 2 m + 2 n)^2 y[m] + (-5 m^2 - 18 m^3 - 17 m^4 + 12 m n + 56 m^2 n + 68 m^3 n - 8 n^2 - 56 m n^2 - 100 m^2 n^2 + 16 n^3 + 64 m n^3 - 16 n^4) y[1 + m] + (1 + m)^2 (-m + 2 n)^2 y[2 + m] == 0, y[0] == 0, y[1] == 1}]][n + 1], {n, 0, 20}] (* Benedict W. J. Irwin, Nov 03 2016 *)

makelist(sum(binomial(2*n-k,k)^2,k,0,n),n,0,20);

{a(n) = sum(k=0, n, binomial(2*n-k, k)^2)} \\ Seiichi Manyama, Jan 13 2019

G.f.: 1/2*(1/sqrt(1-2*sqrt(x)-x-2*x*sqrt(x)+x^2) + 1/sqrt(1+2*sqrt(x)-x+2*x*sqrt(x)+x^2)).
Recurrence: (n-2)*n*(2*n - 1)*(48*n^2 - 192*n + 169)*a(n) = (576*n^5 - 4032*n^4 + 10212*n^3 - 11414*n^2 + 5457*n - 849)*a(n-1) + 5*(2*n - 3)*(48*n^4 - 288*n^3 + 565*n^2 - 399*n + 64)*a(n-2) + (576*n^5 - 4608*n^4 + 13668*n^3 - 18286*n^2 + 10521*n - 1896)*a(n-3) - (n-3)*(n-1)*(2*n - 5)*(48*n^2 - 96*n + 25)*a(n-4). - Vaclav Kotesovec, Mar 02 2014
a(n) ~ phi^(4*n + 2) / (2^(3/2) * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 02 2014, simplified Jan 13 2019
Conjecture: a(n) = hypergeom([-n,-n,n+1,n+1], [1/2,1/2,1], 1/16). - Velin Yanev, Oct 31 2019
a(n) = A051286(2*n). - Mark van Hoeij, Sep 05 2022

A109446 Binomial coefficients C(n,k) with n-k even, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 1, 5, 10, 1, 1, 15, 15, 1, 7, 35, 21, 1, 1, 28, 70, 28, 1, 9, 84, 126, 36, 1, 1, 45, 210, 210, 45, 1, 11, 165, 462, 330, 55, 1, 1, 66, 495, 924, 495, 66, 1, 13, 286, 1287, 1716, 715, 78, 1, 1, 91, 1001, 3003, 3003, 1001, 91, 1, 15, 455, 3003, 6435, 5005

Offset: 0

Philippe Deléham, Aug 27 2005

Binomial(n,2(n-k-1)) is also the number of permutations avoiding both 123 and 132 with k descents, i.e., positions with w[i]>w[i+1]. - Lara Pudwell, Dec 19 2018

			Starred terms in Pascal's triangle (A007318), read by rows:
  1*;
  1,  1*;
  1*,  2,  1*;
  1,  3*,   3,  1*;
  1*,  4,  6*,   4,  1*;
  1,  5*,  10, 10*,   5,   1*;
  1*,  6, 15*,  20, 15*,    6,  1*;
  1,  7*,  21, 35*,  35,  21*,   7,  1*;
  1*,  8, 28*,  56, 70*,   56, 28*,   8, 1*;
  1,  9*,  36, 84*, 126, 126*,  84, 36*,  9, 1*;
Rows in A086645 (1; 1, 1; 1, 6, 1; ...) interspersed with rows in A103327 (1; 3, 1; 5, 10, 1; ...).
1; 1; 1, 1; 3, 1; 1, 6, 1; 5, 10, 1; 1, 15, 15, 1; 7, 35, 21, 1; ....

Alois P. Heinz, Rows n = 0..200, flattened
M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.

Cf. A109447. See A054142 for another version.

T:= (n, k)-> binomial(n, 2*k+irem(n, 2)):
seq(seq(T(n, k), k=0..floor(n/2)), n=0..20);  # Alois P. Heinz, Feb 07 2014

Flatten[ Table[ If[ EvenQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Aug 30 2005

A119274 Triangle of coefficients of numerators in Padé approximation to exp(x).

Original entry on oeis.org

1, 2, 1, 12, 6, 1, 120, 60, 12, 1, 1680, 840, 180, 20, 1, 30240, 15120, 3360, 420, 30, 1, 665280, 332640, 75600, 10080, 840, 42, 1, 17297280, 8648640, 1995840, 277200, 25200, 1512, 56, 1, 518918400, 259459200, 60540480, 8648640, 831600, 55440, 2520

Offset: 0

Paul Barry, May 12 2006

n-th numerator of Padé approximation is (1/n!)*sum{j=0..n, C(n,j)(2n-j)!x^j}. Reversal of A113025. Row sums are A001517. First column is A001813. Inverse is A119275.
Also the Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+2) (A001813) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015
Dividing each diagonal by its initial element generates A054142. - Tom Copeland, Oct 10 2016

			Triangle begins
1,
2, 1,
12, 6, 1,
120, 60, 12, 1,
1680, 840, 180, 20, 1,
30240, 15120, 3360, 420, 30, 1

Cf. A001497, A054142.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (2*n)!/n!, 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[(2#)!/#!&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

# uses[bell_transform from A264428]
# Adds a column 1,0,0,0,... at the left side of the triangle.
def A119274_row(n):
    multifact_4_2 = lambda n: prod(4*k + 2 for k in (0..n-1))
    mfact = [multifact_4_2(k) for k in (0..n)]
    return bell_transform(n, mfact)
[A119274_row(n) for n in (0..9)] # Peter Luschny, Dec 31 2015

Number triangle T(n,k) = C(n,k)(2n-k)!/n!.

After adding a leading column (1,0,0,0,...), the triangle gives the coefficients of the Sheffer associated sequence (binomial-type polynomials) for the delta (lowering) operator D(1-D) with e.g.f. exp[ x * (1 - sqrt(1-4t)) / 2 ] . See Mathworld on Sheffer sequences. See A134685 for relation to Catalan numbers. - Tom Copeland, Feb 09 2008

cp's OEIS Frontend

A116425 Decimal expansion of 2 + 2*cos(2*Pi/7).

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A052535 Expansion of (1-x)*(1+x)/(1-x-2*x^2+x^4).

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A169803 Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).

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A121314 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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A172431 Even row Pascal-square read by antidiagonals.

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A332602 Tridiagonal matrix M read by antidiagonals: main diagonal is 1,2,2,2,2,..., two adjacent diagonals are 1,1,1,1,1,...

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A076756 Triangle of coefficients of characteristic polynomial of M_n, the n X n matrix M_(i,j) = min(i,j).

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A116425 Decimal expansion of 2 + 2cos(2Pi/7).

A052535 Expansion of (1-x)(1+x)/(1-x-2x^2+x^4).