A084534 -id:A084534 - cp's OEIS frontend

A082985 Coefficient table for Chebyshev's U(2n,x) polynomial expanded in decreasing powers of (-4(1-x^2)).

1, 1, 3, 1, 5, 5, 1, 7, 14, 7, 1, 9, 27, 30, 9, 1, 11, 44, 77, 55, 11, 1, 13, 65, 156, 182, 91, 13, 1, 15, 90, 275, 450, 378, 140, 15, 1, 17, 119, 442, 935, 1122, 714, 204, 17, 1, 19, 152, 665, 1729, 2717, 2508, 1254, 285, 19

Offset: 0

Gary W. Adamson, May 29 2003

Sum of row #n = A000204(2n+1), i.e., A002878(n).
Row #n has the unsigned coefficients of a polynomial whose roots are 2 sin(2*Pi*k/(2n+1)) [for k=1 to 2n].
The positive roots are the diagonal lengths of a regular (2n+1)-gon, inscribed in the unit circle.
Polynomial of row #n = Sum_{m=0..n} (-1)^m T(n,m) x^(2n-2m).
This is also the unsigned coefficient table of Chebyshev's 2*T(2*n+1,x) polynomials expanded in decreasing odd powers of 2*x. - Wolfdieter Lang, Mar 07 2007
The n-th row are the coefficients of the polynomial S(n) where S(0)=1, S(1)=x+3, and S(n) = (x+2)*S(n-1) - S(n-2) (see Sun link). - Michel Marcus, Mar 07 2016

			Expansion of polynomials:
  x^0;
  x^2  -  3*x^0;
  x^4  -  5*x^2 +  5*x^0;
  x^6  -  7*x^4 + 14*x^2 -  7*x^0;
  x^8  -  9*x^6 + 27*x^4 - 30*x^2 +  9*x^0;
  x^10 - 11*x^8 + 44*x^6 - 77*x^4 + 55*x^2 - 11*x^0; ...
Polynomial #4 has 8 roots: 2*sin(2*Pi*k/9) for k=1 to 8.
Coefficients (with signs removed) are
  1;
  1,  3;
  1,  5,  5;
  1,  7, 14,  7;
  1,  9, 27, 30,  9;
  1, 11, 44, 77, 55, 11;
  ...

J. D'Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, 1992; see pp. 151,175.
Stephen Eberhart, "Mathematical-Physical Correspondence," Number 37-38, Jan 08, 1982.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

Vincenzo Librandi, Rows n = 0..100 of triangle, flattened
K. Dilcher and K. B. Stolarsky, A Pascal-type triangle characterizing twin primes, Amer. Math. Monthly, 112 (2005), 673-681.
Zhi-Hong Sun, Expansions and identities concerning Lucas sequences, Fibonacci Quart. 44 (2006), no. 2, 145-153. See Theorem 3.1

Cf. companion triangle A084534.
Cf. diagonals: A005408, A000330, A005585, A050486, A054333, A057788.
Cf. A054142, A172431.

Flat(List([0..10], n-> List([0..n], k-> Binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1) ))); # G. C. Greubel, Dec 30 2019

[Binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 30 2019

A082985 := proc(n,m)
    binomial(2*n-m,m)*(2*n+1)/(2*n-2*m+1) ;
end proc: # R. J. Mathar, Sep 08 2013

T[n_, m_]:= Binomial[2*n-m, m]*(2*n+1)/(2*n-2*m+1); Table[T[n, m], {n, 0, 9}, {m, 0, n}]//Flatten (* Jean-François Alcover, Oct 08 2013, after R. J. Mathar *)

T(n,k)=binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1); \\ G. C. Greubel, Dec 30 2019

[[binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 30 2019

Triangle read by rows: row #n has n+1 terms. T(n,0)=1, T(n,n)=2n+1, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k, m-k).
T(k, s) = ((2k+1)/(2s+1))*binomial(k+s, 2s), 0 <= s <= k; then transpose the triangle. - Gary W. Adamson, May 29 2003
From Wolfdieter Lang, Mar 07 2007: (Start)
Signed version: a(n,m)=0 if n < m, otherwise a(n,m) = ((-1)^m)*binomial(2*n+1-m,m)*(2*n+1)/(2*n+1-m). From the Rivlin reference, p. 37, eq.(1.92), using the differential eq. for T(2*n+1,x). Also from Waring's formula.
Signed version: a(n,m)=0 if n < m, otherwise a(n,m) = ((-1)^m)*(Sum_{k=0..n-m} binomial(m+k,k)*binomial(2*n+1,2*(k+m)))/2^(2*(n-m)). Proof: De Moivre's formula for cos((2*n+1)*phi) rewritten in terms of odd powers of cos(phi). Cf. Rivlin reference p. 4, eq.(1.10).
Signed version: a(n,m) = A084930(n,n-m)/2^(2*(n-m)) (scaled coefficients of Chebyshev's T(2*n+1,x), decreasing odd powers).
Unsigned version: a(n,m)=0 if n < m, otherwise a(n,m) = binomial(2*n-m,m)*(2*n+1)/(2*(n-m)+1). From the differential eq. for U(2*n,x). (End)
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2). - Philippe Deléham, Feb 24 2012
Sum_{i>=0} T(n-i,n-2*i) = A003945(n). - Philippe Deléham, Feb 24 2012
Sum_{i>=0} T(n-i, n-2*i)*4^i = 3^n = A000244(n). - Philippe Deléham, Feb 24 2012
From Paul Weisenhorn Nov 25 2019: (Start)
T(r,k) = binomial(2*r-k,k-1) + binomial(2*r-1-k,k-2) with 1 <= r and 1 <= k <= r.
For a given n, one gets r = floor((1+sqrt(8*n))/2), k = n-(r^2-r)/2, a(n) = binomial(2*r-k,k-1) + binomial(2*r-1-k,k-2). (End)

Edited by Anne Donovan (anned3005(AT)aol.com), Jun 11 2003

Re-edited by Don Reble, Nov 12 2005

A185828 Half the number of n X 2 binary arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

Original entry on oeis.org

1, 3, 10, 23, 61, 162, 421, 1103, 2890, 7563, 19801, 51842, 135721, 355323, 930250, 2435423, 6376021, 16692642, 43701901, 114413063, 299537290, 784198803, 2053059121, 5374978562, 14071876561, 36840651123, 96450076810, 252509579303

Offset: 1

R. H. Hardin, Feb 05 2011

nonn

Column 2 of A185835.

			Some solutions for 4 X 2 with a(1,1)=0:
  0 0   0 1   0 0   0 0   0 1   0 0   0 0   0 0   0 0   0 0
  1 1   0 1   0 1   1 1   0 1   1 0   0 1   1 1   1 0   0 1
  0 1   0 0   0 1   0 1   1 0   1 0   1 1   1 1   1 1   0 1
  0 0   1 1   0 0   0 1   1 0   0 0   0 0   0 0   0 0   0 1
The logarithmic g.f. begins:
L(x) = x + 3*x^2/2 + 10*x^3/3 + 23*x^4/4 + 61*x^5/5 + 162*x^6/6 + ..., where
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 26*x^5 + 63*x^6 + ... + A051286(n)*x^n/n + ... - _Paul D. Hanna_, Mar 19 2011

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A051286 (exp), A180662 (Fi1).

a := proc(n): n*add(binomial(2*n-2*k, 2*k)/(n-k), k=0..n-1) end: seq(a(n), n=1..28); # Johannes W. Meijer, Jun 18 2018

{a(n)=n*sum(k=0, n-1, binomial(2*n-2*k, 2*k)/(n-k))} /* Paul D. Hanna, Mar 19 2011 */

{a(n)=n*polcoeff(-log( (1+x+x^2)*(1-3*x+x^2) +x*O(x^n))/2, n)} /* Paul D. Hanna, Mar 19 2011 */

Empirical: a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3) - a(n-4).
a(n) = n*Sum_{k=0..n-1} C(2n-2k, 2k)/(n-k). - Paul D. Hanna, Mar 19 2011
L.g.f.: Sum_{n>=1} a(n)*x^n/n = -log((1+x+x^2)*(1-3*x+x^2))/2. - Paul D. Hanna, Mar 19 2011
Logarithmic derivative of A051286, which is the Whitney number of level n of the lattice of the ideals of the fence of order 2n. - Paul D. Hanna, Mar 19 2011
Empirical g.f.: x*(1+x+3*x^2-2*x^3)/(1+x+x^2)/(1-3*x+x^2). - Colin Barker, Feb 22 2012
Empirical: a(n) = Sum_{k=0..floor(n/2)} A084534(n, 2*k). - Johannes W. Meijer, Jun 17 2018
Empirical: a(n) = A100886(2n). - Wojciech Florek, Jan 26 2020

A049853 a(n) = a(n-1) + Sum_{k=0..n-3} a(k) for n >= 2, a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 2, 3, 6, 11, 19, 33, 58, 102, 179, 314, 551, 967, 1697, 2978, 5226, 9171, 16094, 28243, 49563, 86977, 152634, 267854, 470051, 824882, 1447567, 2540303, 4457921, 7823106, 13728594, 24092003, 42278518, 74193627

Offset: 0

Clark Kimberling

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1).

Cf. A070550, A180662 (Ca2).

a049853 n = a049853_list !! n
a049853_list = 1 : 2 : 2 : 3 :
   zipWith (+) a049853_list
               (zipWith (+) (drop 2 a049853_list) (drop 3 a049853_list))
-- Reinhard Zumkeller, Aug 06 2011

a := proc(n) option remember: if n<2 then n+1 else a(n-1) + add(a(k), k=0..n-3) fi end: seq(a(n), n=0..33); # Johannes W. Meijer, Jun 18 2018

LinearRecurrence[{2,-1,1},{1,2,2},40] (* Harvey P. Dale, May 12 2022 *)

Vec((1 - x)*(1 + x) / (1 - 2*x + x^2 - x^3) + O(x^40)) \\ Colin Barker, Jun 17 2018

a(n) = 2*a(n-1) - a(n-2) + a(n-3); 3 initial terms required.
a(n) = a(n-1) + a(n-2) + a(n-4) for n > 3. - Reinhard Zumkeller, Aug 06 2011
Empirical: a(n) = Sum_{k=0..floor(n/3)} A084534(n-2*k, n-3*k). - Johannes W. Meijer, Jun 17 2018
G.f.: (1 - x)*(1 + x) / (1 - 2*x + x^2 - x^3). - Colin Barker, Jun 17 2018

A158982 Coefficients of polynomials P(n,x):=-2+P(n-1,x)^2, where P(0,x)=x-2.

Original entry on oeis.org

1, -2, 1, -4, 2, 1, -8, 20, -16, 2, 1, -16, 104, -352, 660, -672, 336, -64, 2, 1, -32, 464, -4032, 23400, -95680, 283360, -615296, 980628, -1136960, 940576, -537472, 201552, -45696, 5440, -256, 2, 1, -64, 1952, -37760, 520144, -5430656, 44662464

Offset: 1

Clark Kimberling, Apr 02 2009

The 2^n zeros of P(n,x) are 2+2*cos[(2k-1)Pi/(2^(n+1))], k=1,2,...,2^n.

P(n,x) = 2*T(2^(n+1),(1/2)x^(1/2)), where T(k,t) is the k-th Chebyshev polynomial of the first kind.

			Row 1: 1 -2 (from x-2)
Row 2: 1 -4 2 (from x^2-4x+2)
Row 3: 1 -8 20 -16 2
Row 4: 1 -16 104 -352 660 -672 336 -64 2

Clark Kimberling, Polynomials defined by a second-order recurrence, interlacing zeros, and Gray codes, The Fibonacci Quarterly 48 (2010) 209-218.

Cf. A084534, A158983, A158984, A158985, A158986.

tabf(nn) = {p = x-2; print(Vec(p)); for (n=2, nn, p = -2 + p^2; print(Vec(p)););} \\ Michel Marcus, Mar 01 2016

P(n+1,x+2) = P(n,x^2) for n>=0.

A118007 Triangle, diagonals generated from Lucas polynomials.

Original entry on oeis.org

2, 3, 2, 7, 4, 2, 18, 14, 5, 2, 47, 52, 23, 6, 2, 123, 194, 110, 34, 7, 2, 322, 724, 527, 198, 47, 8, 2, 843, 2702, 2525, 1154, 322, 62, 9, 2, 2207, 10084, 12098, 6726, 2207, 488, 79, 10, 2, 5778, 37634, 57965, 39202, 15127, 3842, 702, 98, 11, 2

Offset: 0

Gary W. Adamson, Apr 09 2006

Leftmost column = A005248, bisection of Lucas sequence A000032.

Refer to A084534 for a variation of the Lucas polynomials.

			First few rows of the triangle:
    2;
    3,   2;
    7,   4,   2;
   18,  14,   5,  2;
   47,  52,  23,  6, 2;
  123, 194, 110, 34, 7, 2;
  ...
For example, 4th diagonal from the right (18, 52, 110, ...) = f(x), x=1,2,3, ...: x^3 + 6x^2 + 9x + 2.
(18, 52, 110, ...) = binomial transform of 4th row of A118008: (18, 34, 24, 6).

Jay Kappraff, "Beyond Measure, A Guided tour Through Nature, Myth and Number", World Scientific, 2002, p. 485 (Table 22.6b).

Cf. A000032, A005248, A034807, A118008.

Cf. A084534.

TLucas(n,k) = binomial(n-k, k) + binomial(n-k-1, k-1) + (n==0); \\ A084534
pol(n) = Pol(vector(n+1, k, TLucas(2*n,k-1)));
T(n,k) = subst(pol(n-k), x, k+1);
trgT(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Aug 12 2022

Diagonals are sequences as f(x), x=1,2,3; Lucas polynomials in the format: (2); (x + 2); (x^2 + 4x + 2); (x^3 + 6x^2 + 9x + 2); (x^4 + 8x^3 + 20x^2 + 16x + 2); (x^5 + 10x^4 + 35x^3 + 50x^2 + 25x + 2); ...

Diagonals of the triangle are binomial transforms of A118008 rows.

More terms from Michel Marcus, Aug 12 2022

A302676 Number of n X 4 0..1 arrays with every element equal to 0, 1, 4 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

Original entry on oeis.org

5, 5, 12, 20, 33, 64, 121, 231, 440, 838, 1597, 3042, 5796, 11042, 21037, 40079, 76357, 145473, 277150, 528017, 1005960, 1916521, 3651291, 6956316, 13252938, 25249049, 48103634, 91645416, 174599746, 332641529, 633737387, 1207375029, 2300250057, 4382358586

Offset: 1

R. H. Hardin, Apr 11 2018

nonn

Column 4 of A302680.

Empirical: The antidiagonal sums of A084534 lead to the terms of this sequence for n >= 5. - Johannes W. Meijer, Jun 17 2018

			Some solutions for n=5:
  0 1 0 1     0 1 0 1     0 0 0 1     0 1 0 1     0 1 1 1
  0 1 0 1     0 0 0 1     0 1 0 1     0 1 0 1     0 1 0 1
  0 1 0 1     0 1 0 1     0 1 0 1     0 0 0 1     0 1 0 1
  0 1 0 1     0 1 1 1     0 1 1 1     0 1 0 1     0 1 0 1
  0 1 0 1     0 1 0 1     0 1 0 1     0 1 1 1     0 0 0 1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A302680, A180662 (Kn11), A084534.

Empirical: a(n) = a(n-1) + 2*a(n-2) - a(n-4) for  n > 8.
Empirical g.f.: x*(5 - 3*x^2 - 2*x^3 - 6*x^4 - 4*x^5 + 3*x^6 + 2*x^7) / (1 - x - 2*x^2 + x^4). - Colin Barker, Jun 17 2018
The data in the range n = 6..210 is matched by h(n) = hypergeom([-n+1, -(1/2)*n, 1/4-(1/2)*n, -(1/2)*n+1/2, -(1/2)*n+3/4], [-n, -(2/3)*n+1, -(2/3)*n+2/3, -(2/3)*n+1/3], -256/27). - Peter Luschny, Aug 24 2018

A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)(1+1/(sqrt(1-u^2)))exp(p*sqrt(1-u^2)).

Original entry on oeis.org

1, 1, -2, 3, -4, 2, 15, -18, 9, -2, 105, -120, 60, -16, 2, 945, -1050, 525, -150, 25, -2, 10395, -11340, 5670, -1680, 315, -36, 2, 135135, -145530, 72765, -22050, 4410, -588, 49, -2, 2027025, -2162160, 1081080, -332640, 69300, -10080, 1008, -64, 2

Offset: 0

Johannes W. Meijer, May 31 2018

The function f(u, p) = (1/2)*(1+1/(sqrt(1-u^2))) * exp(p*sqrt(1-u^2)) was found while studying the Fresnel-Kirchhoff and the Rayleigh-Sommerfeld theories of diffraction, see the Meijer link.
The Taylor expansion of f(u, p) leads to the number triangle T(n, k), see the example section.
Normalization of the triangle terms, dividing the T(n, k) by T(n-k, 0), leads to A084534.
The row sums equal A003436, n >= 2, respectively A231622, n >= 1.

			The first few terms of the Taylor expansion of f(u; p) are:
f(u, p) = exp(p) * (1 + (1-2*p) * u^2/4 + (3-4*p+2*p^2) * u^4/16 + (15-18*p+9*p^2-2*p^3) * u^6/96 + (105-120*p+60*p^2-16*p^3+2*p^4) * u^8/768 + ... )
The first few rows of the T(n, k) triangle are:
n=0:     1
n=1:     1,     -2
n=2:     3,     -4,    2
n=3:    15,    -18,    9,    -2
n=4:   105,   -120,   60,   -16,   2
n=5:   945,  -1050,  525,  -150,  25,  -2
n=6: 10395, -11340, 5670, -1680, 315, -36, 2

J. W. Goodman, Introduction to Fourier Optics, 1996.
A. Papoulis, Systems and Transforms with Applications in Optics, 1968.

Andrew Howroyd, Rows n=0..50 of triangle, flattened
M. J. Bastiaans, The Wigner distribution function applied to optical signals and systems, Optics Communications, Vol. 25, nr. 1, pp. 26-30, 1978.
H. J. Butterweck, General theory of linear, coherent optical data processing systems, Journal of the Optical Society of America, Vol. 67, nr. 1, pp. 60-70, 1977.
J. W. Meijer, A note on optical diffraction, 1979.

Cf. A003436, A231622, A032184, A084534, A127674, A132062, A001497.
Cf. Related to the left hand columns: A001147, A001193, A261065.
Cf. Related to the right hand columns: A280560, A162395, A006011, A040977, A053347, A054334, A266561.

[[n le 0 select 1 else (-1)^k*2^(k-n+1)*Factorial(2*n-k-1)*Binomial(n, k)/Factorial(n-1): k in [0..n]]: n in [1..10]]; // G. C. Greubel, Nov 08 2018

T := proc(n, k): if n=0 then 1 else (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!) fi: end: seq(seq(T(n, k), k=0..n), n=0..8);

Table[If[n==0 && k==0,1, (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!)], {n, 0, 10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 08 2018 *)

T(n,k) = {if(n==0, 1, (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 08 2018

T(n, k) = (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!), n > 0 and 0 <= k <= n, T(0, 0) = 1.
T(n, k) = (-1)^k*A001147(n-k)*A084534(n, k), n >= 0 and 0 <= k <= n.
T(n, k) = 2^(2*(k-n)+1)*A001147(n-k)*A127674(n, n-k), n > 0 and 0 <= k <= n, T(0, 0) = 1.
T(n, k) = (-1)^k*(A001497(n, k) + A132062(n, k)), n >= 1, T(0,0) = 1.

cp's OEIS Frontend

A082985 Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).

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A185828 Half the number of n X 2 binary arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

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A049853 a(n) = a(n-1) + Sum_{k=0..n-3} a(k) for n >= 2, a(0)=1, a(1)=2.

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A158982 Coefficients of polynomials P(n,x):=-2+P(n-1,x)^2, where P(0,x)=x-2.

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A118007 Triangle, diagonals generated from Lucas polynomials.

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A302676 Number of n X 4 0..1 arrays with every element equal to 0, 1, 4 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

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A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)*(1+1/(sqrt(1-u^2)))*exp(p*sqrt(1-u^2)).

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A082985 Coefficient table for Chebyshev's U(2n,x) polynomial expanded in decreasing powers of (-4(1-x^2)).

A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)(1+1/(sqrt(1-u^2)))exp(p*sqrt(1-u^2)).