cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A054332 One half of tenth unsigned column of Lanczos triangle A053125 (decreasing powers).

Original entry on oeis.org

5, 440, 16016, 366080, 6223360, 85995520, 1018716160, 10711072768, 102385254400, 905301196800, 7501067059200, 58822597017600, 439993025691648, 3158924287016960, 21879051958353920, 146801380881858560
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

References

  • 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.

Links

Crossrefs

Cf. A053125, A054328.

Programs

  • GAP
    List([0..20], n-> 2^(2*n-1)*Binomial(2*n+10, 9)); # G. C. Greubel, Jul 22 2019
  • Magma
    [2^(2*n-1)*Binomial(2*n+10, 9): n in [0..20]]; // G. C. Greubel, Jul 22 2019
  • Mathematica
    Table[2^(2*n-1)*Binomial[2*n+10,9], {n,0,20}] (* G. C. Greubel, Jul 22 2019 *)
  • PARI
    vector(20, n, n--; 2^(2*n-1)*binomial(2*n+10, 9)) \\ G. C. Greubel, Jul 22 2019
  • Sage
    [2^(2*n-1)*binomial(2*n+10, 9) for n in (0..20)] # G. C. Greubel, Jul 22 2019

Formula

a(n) = 2^(2*n-1)*binomial(2*n+10, 9) = -A053125(n+9, 9)/2 = A054328(n)/2.
G.f.: (1+40*x+80*x^2)*(5+40*x+16*x^2)/(1-4*x)^10.

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

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, 1, -20, 171, -816, 2380, -4368, 5005, -3432, 1287, -220, 11, 1
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

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

Examples

			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.

References

  • 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.

Links

Crossrefs

Cf. A053124, A053122, A172431.

Programs

  • GAP
    Flat(List([0..10], n-> List([0..n], k-> (-1)^k*Binomial(2*n-k+1,k) ))); # G. C. Greubel, Jul 23 2019
  • Magma
    [(-1)^k*Binomial(2*n-k+1,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 23 2019
  • Maple
    A053123 := proc(n,m)
    (-1)^m*binomial(2*n+1-m,m) ;
end proc: # R. J. Mathar, Sep 08 2013
  • Mathematica
    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 *)
  • PARI
    for(n=0,10, for(k=0,n, print1((-1)^k*binomial(2*n-k+1,k), ", "))) \\ G. C. Greubel, Jul 23 2019
  • Sage
    [[(-1)^k*binomial(2*n-k+1,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 23 2019

Formula

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

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

  • 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).

Links

Crossrefs

Found in A053125 and A053124.
Cf. A000051, A002457, A002697, A212698.

Programs

Formula

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

A053124 Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in increasing order).

Original entry on oeis.org

1, -2, 4, 3, -16, 16, -4, 40, -96, 64, 5, -80, 336, -512, 256, -6, 140, -896, 2304, -2560, 1024, 7, -224, 2016, -7680, 14080, -12288, 4096, -8, 336, -4032, 21120, -56320, 79872, -57344, 16384, 9, -480, 7392, -50688, 183040, -372736, 430080, -262144, 65536, -10, 660, -12672, 109824, -512512, 1397760, -2293760, 2228224
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

a(n,m) = (4^m)*A053122(n,m).
G.f. for row polynomials U^{*}(n,x) = U(n,2*x-1) (signed triangle): 1/(1+2*z*(1-2*x) + z^2). Unsigned triangle |a(n,m)| has g.f. 1/(1-2*z*(1+2*x)+z^2) for the row polynomials.
Row sums (signed triangle) A000027(n+1) (natural numbers). Row sums (unsigned triangle) A001109(n+1).
In the language of Shapiro et al. (see A053121 for the reference) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to a Riordan group.

Examples

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

References

  • C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
  • G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 1, problem 39, page 7.
  • Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

Links

Crossrefs

Cf. A053122, A053125.

Programs

  • Mathematica
    Table[ CoefficientList[ ChebyshevU[n, 2x - 1], x], {n, 0, 9}] // Flatten (* Jean-François Alcover, Dec 05 2012 *)

Formula

a(n, m) := 0 if n < m, otherwise (4^m)*((-1)^(n-m))*binomial(n+m+1, 2*m+1);
a(n, m) = -2*a(n-1, m) + 4*a(n-1, m-1) - a(n-2, m), a(n, m) := 0 if n=-1 or m=-1 or n < m, a(0, 0)=1;
g.f. for m-th column (signed triangle): ((4*x/(1+x)^2)^m)/(1+x)^2.
In other words, Riordan array (1/(1+x)^2, 4x/(1+x)^2). - Ralf Stephan, Jan 21 2014
Previous Showing 11-14 of 14 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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