login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A084534 Triangle read by rows: row #n has n+1 terms. T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k). 7
1, 1, 2, 1, 4, 2, 1, 6, 9, 2, 1, 8, 20, 16, 2, 1, 10, 35, 50, 25, 2, 1, 12, 54, 112, 105, 36, 2, 1, 14, 77, 210, 294, 196, 49, 2, 1, 16, 104, 352, 660, 672, 336, 64, 2, 1, 18, 135, 546, 1287, 1782, 1386, 540, 81, 2, 1, 20, 170, 800, 2275, 4004, 4290, 2640, 825, 100, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Sum of row #n = A000204(2n). (But sum of row #0 = 1.)

Row #n has the unsigned coefficients of the monic polynomial whose roots are 2 cos(Pi*(2k-1)/(4n)) for k=1..2n. [Comment corrected by Barry Brent, Jan 03 2006]

The positive roots are some diagonal lengths of a regular (4n)-gon, inscribed in the unit circle.

Polynomial of row #n = Sum_{m=0..n} [(-1)^m] T(n,m) x^(2n-2m).

This is the unsigned version of the coefficient table for scaled Chebyshev T(2*n,x) polynomials. - Wolfdieter Lang, Mar 07 2007

Reversed A127677 (cf. A156308, A217476, A263916). - Tom Copeland, Nov 07 2015

REFERENCES

I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124. See p. 118.

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. p. 37, eq.(1.96) and p. 4. eq.(1.10).

LINKS

Table of n, a(n) for n=0..65.

I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124. [Scan of annotated copy]

FORMULA

T(n,m) = binomial(2*n-m, m)*2*n/(2*n-m) for n > 0. - Andrew Howroyd, Dec 18 2017

Signed version: a(n,m)=0 if n<m, a(0,0)=1 else a(n,m)=((-1)^m)*binomial(2*n-m,m)*2*n/(2*n-m). - Wolfdieter Lang, Mar 07 2007

Signed version: a(n,m)=0 if n<m, a(0,0)=1 else a(n,m)=((-1)^m)*Sum_{l=0..n-m} binomial(m+l,l)*binomial(2*n,2*(l+m))/2^(2*(n-m)-1). - Wolfdieter Lang, Mar 07 2007

Signed version: a(n,m)=0 if n<m, a(0,0)=1 else a(n,m)= A127674(n,n-m)/2^(2*(n-m)-1) (scaled coefficients of Chebyshev's T(2*n,x)), decreasing even powers). - Wolfdieter Lang, Mar 07 2007; corrected by Johannes W. Meijer, May 31 2018

EXAMPLE

1

x^2 - 2

x^4 - 4x^2 + 2

x^6 - 6x^4 + 9x^2 - 2

x^8 - 8x^6 + 20x^4 - 16x^2 + 2

x^10 - 10x^8 + 35x^6 - 50x^4 + 25x^2 - 2

Polynomial #4 has 8 roots: 2 sin(Pi*k/16) for k=1,3,5,...,15.

MAPLE

T := proc(n, m): if n=0 then 1 else binomial(2*n-m, m)*2*n/(2*n-m) fi: end: seq(seq(T(n, m), m=0..n), n=0..10); # Johannes W. Meijer, May 31 2018

MATHEMATICA

a[n_, m_] := Binomial[2n-m, m]*2n/(2n-m); a[0, 0] = 1; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Apr 12 2016, after Wolfdieter Lang *)

PROG

(PARI) T(n, m) = if(n==0, m==0, binomial(2*n-m, m)*2*n/(2*n-m)) \\ Andrew Howroyd, Dec 18 2017

CROSSREFS

Row sums are A005248 for n > 0.

Companion triangle A082985.

Cf. A082985 (unsigned scaled coefficient table for Chebyshev's T(2*n+1, x) polynomials).

Cf. A127677, A156308, A217476, A263916.

Sequence in context: A304223 A035607 A059370 * A165899 A316354 A104582

Adjacent sequences:  A084531 A084532 A084533 * A084535 A084536 A084537

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, May 29 2003

EXTENSIONS

Edited by Don Reble, Nov 12 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 13 04:49 EDT 2021. Contains 344980 sequences. (Running on oeis4.)