A053124 -id:A053124 - cp's OEIS frontend

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, 1, -20, 171, -816, 2380, -4368, 5005, -3432, 1287, -220, 11, 1

Offset: 0

Wolfdieter Lang

T(n,m) = A053122(n,n-m).
G.f. for row polynomials and row sums same as in A053122.
Unsigned column sequences are A000012, A005843, A014105, A002492 for m=0..3, resp. and A053126-A053131 for m=4..9.
This is also the coefficient triangle for Chebyshev's U(2*n+1,x) polynomials expanded in decreasing odd powers of (2*x): U(2*n+1,x) = Sum_{m=0..n} T(n,m)*(2*x)^(2*(n-m)+1). See the W. Lang link given in A053125.
Unsigned version is mirror image of A078812. - Philippe Deléham, Dec 02 2008

			Triangle 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; ...
E.g. fourth row (n=3) {1,-6,10,-4} corresponds to polynomial S(3,x-2) = x^3-6*x^2+10*x-4.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
Stephen Barnett, "Matrices: Methods and Applications", Oxford University Press, 1990, p. 132, 343.

T. D. Noe, Rows n=0..50 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for sequences related to Chebyshev polynomials.

Cf. A053124, A053122, A172431.

Flat(List([0..10], n-> List([0..n], k-> (-1)^k*Binomial(2*n-k+1,k) ))); # G. C. Greubel, Jul 23 2019

[(-1)^k*Binomial(2*n-k+1,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 23 2019

A053123 := proc(n,m)
    (-1)^m*binomial(2*n+1-m,m) ;
end proc: # R. J. Mathar, Sep 08 2013

T[n_, m_]:= (-1)^m*Binomial[2*n+1-m, m]; Table[T[n, m], {n, 0, 11}, {m, 0, n}]//Flatten (* Jean-François Alcover, Mar 05 2014, after R. J. Mathar *)

for(n=0,10, for(k=0,n, print1((-1)^k*binomial(2*n-k+1,k), ", "))) \\ G. C. Greubel, Jul 23 2019

[[(-1)^k*binomial(2*n-k+1,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 23 2019

T(n, m) = 0 if n

T(n, m) = -2*T(n-1, m-1) + T(n-1, m) - T(n-2, m-2), T(n, -2) = 0, T(-2, m) = 0, T(n, -1) = 0 = T(-1, m), T(0, 0) = 1, T(n, m) = 0 if n

G.f. for m-th column (signed triangle): ((-1)^m)*x^m*Po(m+1, x)/(1-x)^(m+1), with Po(k, x) := Sum_{j=0..floor(k/2)} binomial(k, 2*j+1)*x^j.

The n-th degree polynomial is the characteristic equation for an n X n tridiagonal matrix with (diagonal = all 2's, sub and superdiagonals all -1's and the rest 0's), exemplified by the 4X4 matrix M = [2 -1 0 0 / -1 2 -1 0 / 0 -1 2 -1 / 0 0 -1 2]. - Gary W. Adamson, Jan 05 2005

Sum_{m=0..n} T(n,m)*(c(n))^(2*n-2*m) = 1/c(n), where c(n) = 2*cos(Pi/(2*n+3)). - L. Edson Jeffery, Sep 13 2013

A053125 Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in decreasing order).

Original entry on oeis.org

1, 4, -2, 16, -16, 3, 64, -96, 40, -4, 256, -512, 336, -80, 5, 1024, -2560, 2304, -896, 140, -6, 4096, -12288, 14080, -7680, 2016, -224, 7, 16384, -57344, 79872, -56320, 21120, -4032, 336, -8, 65536, -262144, 430080, -372736, 183040, -50688, 7392, -480, 9, 262144, -1179648, 2228224, -2293760, 1397760

Offset: 0

Wolfdieter Lang

A000302 (powers of 4), A002699, A002700 unsigned column sequences for m=0..2.
G.f. for row polynomials U(n,2*x-1) and row sums same as for A053124.
With offset 1 this is also the coefficient triangle of 2* U(2*n-1,x) expanded in decreasing powers of x. W. Lang, Mar 07 2007.

			{1}; {4,-2}; {16,-16,3}; {64,-96,40,-4}; {256,-512,336,-80,5};... E.g. fourth row (n=3) corresponds to polynomial U^{*}(3,m)=U(3,2*x-1)= 64*x^3-96*x^2+40*x-4.

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

T. D. Noe, Rows n=0..50 of triangle, flattened
W. Lang, First rows and related triangles .
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind
Index entries for sequences related to Chebyshev polynomials.

Cf. A053124, A053123.

Reverse /@ CoefficientList[Table[ChebyshevU[n, 2 x - 1], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 04 2018 *)
Reverse /@ CoefficientList[ChebyshevU[Range[0, 10], 2 x - 1], x] // Flatten (* Eric W. Weisstein, Apr 04 2018 *)

a(n, m) = A053124(n, n-m)= (4^(n-m))*A053123(n, m)= (4^(n-m))*((-1)^m)*binomial(2*n+1-m, m) if n >= m, else 0.

a(n, m) := -2*a(n-1, m-1)+4*a(n-1, m)-a(n-2, m-2), a(-2, m) := 0=: a(n, -2), a(-1, m) := 0=: a(n, -1), a(0, 0)=1, a(n, m)=0 if n

G.f. for m-th column (signed triangle): ((-x)^m)*Po(m+1, 4*x)/(1-4*x)^(m+1), with Po(k, x) := sum('binomial(k, 2*j+1)*x^j', 'j'=0..floor(k/2)).

A002699 a(n) = n2^(2n-1).

Original entry on oeis.org

0, 2, 16, 96, 512, 2560, 12288, 57344, 262144, 1179648, 5242880, 23068672, 100663296, 436207616, 1879048192, 8053063680, 34359738368, 146028888064, 618475290624, 2611340115968, 10995116277760, 46179488366592, 193514046488576

Offset: 0

N. J. A. Sloane

Right side of binomial sum Sum(i * binomial(2*n, i), i=1..n) - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
Coefficients of shifted Chebyshev polynomials.
Starting with offset 1 = 4th binomial transform of [2, 8, 0, 0, 0, ...]. - Gary W. Adamson, Jul 21 2009
Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then a(n) = the sum of the size of the symmetric difference of x and y for every (x,y) of B. - Ross La Haye, Jan 04 2013
It's the relation [27] with T(n) in the document of Ross. Following the last comment of Ross, A002697 is the similar sequence when replacing "symmetric difference" by "intersection" and A212698 is the similar  sequence when replacing "symmetric difference" by union. - Bernard Schott, Jan 04 2013
If Delta = Symmetric difference, here, X Delta Y and Y Delta X are considered as two distinct Cartesian products, if we want to consider that X Delta Y = X Delta Y is the same Cartesian product, see A002697. - Bernard Schott, Jan 15 2013

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Rebecca Bourn and William Q. Erickson, A palindromic polynomial connecting the earth mover's distance to minuscule lattices of Type A, arXiv:2307.02652 [math.CO], 2023.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Found in A053125 and A053124.

Cf. A000051, A002457, A002697, A212698.

[n*2^(2*n-1): n in [0..30]]; /* or */ I:=[0, 2]; [n le 2 select I[n] else 8*Self(n-1)-16*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 20 2013

A002699 := n->n*2^(2*n-1);
A002699:=2*z/(4*z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

Table[(n 2^(2 n - 1)), {n, 0, 30}] (* Vincenzo Librandi, Mar 20 2013 *)
LinearRecurrence[{8,-16},{0,2},30] (* Harvey P. Dale, Dec 20 2015 *)

a(n)=n*2^(2*n-1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2 * A002697(n). - Bernard Schott, Jan 04 2013
a(n) = A212698(n) - A002697(n)
a(n) = 8*a(n-1)-16*a(n-2) with n>1, a(0)=0, a(1)=2. - Vincenzo Librandi, Mar 20 2013
G.f.: (2*x)/(1 - 4*x)^2. - Harvey P. Dale, Jul 28 2021
E.g.f.: (exp(4*x) - 1)/2. - Stefano Spezia, Aug 04 2022

A091044 One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.

Original entry on oeis.org

1, 2, 2, 3, 10, 3, 4, 28, 28, 4, 5, 60, 126, 60, 5, 6, 110, 396, 396, 110, 6, 7, 182, 1001, 1716, 1001, 182, 7, 8, 280, 2184, 5720, 5720, 2184, 280, 8, 9, 408, 4284, 15912, 24310, 15912, 4284, 408, 9, 10, 570, 7752, 38760, 83980, 83980, 38760, 7752, 570, 10, 11

Offset: 1

Wolfdieter Lang, Jan 23 2004

The odd-numbered columns of this triangle can be reduced: see triangle A091043.
The odd-numbered rows coincide with the ones of the reduced triangle A091043.
binomial(2*n,2*m+1) is even for n >= m + 1 >= 1, hence every T(n,m) is a positive integer.
The GCD (greatest common divisor) of the entries of each odd-numbered row n=2*k+1, k>=0, is 1.
The GCD of the entries of the even-numbered row n=2*k, k>=1, is A006519(n) (highest power of 2 in n=2*k).

			Triangle begins:
  [1];
  [2,2];
  [3,10,3];
  [4,28,28,4];
  [5,60,126,60,5];
  [6,110,396,396,110,6];
  ...
n = 6 = 2*3: gcd(6,110,396) = 2 = A006519(6);
n = 5: gcd(5,60,126) = 1 = A006519(5).

Indranil Ghosh, Rows 1..125, flattened
Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Honeycombs in the Pascal triangle and beyond, arXiv:2203.13205 [math.HO], 2022. See p. 4.
Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, and André Schulz, The Number of Convex Polyominoes with Given Height and Width, arXiv:1903.01095 [math.CO], 2019.
Hernan de Alba, W. Carballosa, J. Leaños, and L. M. Rivera, Independence and matching numbers of some token graphs, arXiv preprint arXiv:1606.06370 [math.CO], 2016.
Wolfdieter Lang, First 9 rows.

Cf. A000302 (row sums), A001109, A006519, A007318, A053124, A091042, A091043, A111910.

Flatten[Table[Binomial[2n,2m+1]/2,{n,1,11},{m,0,n-1}]] (* Indranil Ghosh, Feb 22 2017 *)

{A(i, j) = binomial(2*i + 2*j - 2, 2*i - 1) / 2}; /* Michael Somos, Oct 15 2017 */

T(n, m)= binomial(2*n, 2*m+1)/2, n >= m + 1 >= 1, else 0.
  Put a(n) = n!*(n+1/2)!/(1/2)!. T(n+1,k) = (n+1)*a(n)/(a(k)*a(n-k)).
  T(n-1,k-1)*T(n,k+1)*T(n+1,k) = T(n-1,k)*T(n,k-1)*T(n+1,k+1). Cf. A111910. - Peter Bala, Oct 13 2011
From Peter Bala, Jul 29 2013: (Start)
O.g.f.: 1/(1 - 2*t*(x + 1) + t^2*(x - 1)^2)= 1 + (2 + 2*x)*t + (3 + 10*x + 3*x^2)*t^2 + ....
The n-th row polynomial R(n,x) = 1/(4*sqrt(x))*( (1 + sqrt(x))^(2*n) - (sqrt(x) - 1)^(2*n) ) and has n-1 real zeros given by the formula -cot^2(k*Pi/(2*n)) for k = 1,2,...,n-1. Cf A091042.
The row polynomial R(n,x) satisfies (x - 1)^n*R(n,x/(x - 1)) = U(n,2*x - 1), the n-th row polynomial of A053124.
Row sums A000302. Sum {k = 0..n-1} 2^k*T(n,k) = A001109(n). (End)
From Werner Schulte, Jan 13 2017: (Start)
(1) T(n,m) = T(n-1,m) + T(n-1,m-1)*(2*n-1-m)/m for 0 < m < n-1 with T(n,0) = n and T(n,n) = 0;
(2) T(n,m) = 2*T(n-1,m) + 2*T(n-1,m-1) - T(n-2,m) + 2*T(n-2,m-1) - T(n-2,m-2) for 0 < m < n-1 with T(n,0) = T(n,n-1) = n and T(n,m) = 0 if m < 0 or m >= n;
(3) The row polynomials p(n,x) = Sum_{m=0..n-1} T(n,m)*x^m satisfy the recurrence equation p(n+2,x) = (2+2*x)*p(n+1,x) - (x-1)^2*p(n,x) for n >= 1 with initial values p(1,x) = 1 and p(2,x) = 2+2*x.
(End)
G.f.: x*y /(1 - 2*(x+y) + (x-y)^2) with the entries regarded as an infinite square array A(i, j) read by antidiagonals. - Michael Somos, Oct 15 2017

Showing 1-4 of 4 results.

cp's OEIS Frontend

A053123 Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A053125 Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in decreasing order).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A002699 a(n) = n*2^(2*n-1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A091044 One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A002699 a(n) = n2^(2n-1).