A084534 - OEIS

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).

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^(2*n-2*m).

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

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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 from Wolfdieter Lang, Mar 07 2007: (Start)

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).

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).

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). [corrected by Johannes W. Meijer, May 31 2018] (End)

EXAMPLE

First few Chebyshev T(2*n,x) polynomials:

T(2*0,x) = 1;

T(2*1,x) = x^2 - 2;

T(2*2,x) = x^4 - 4*x^2 + 2;

T(2*3,x) = x^6 - 6*x^4 + 9*x^2 - 2;

T(2*4,x) = x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 2;

T(2*5,x) = x^10 - 10*x^8 + 35*x^6 - 50*x^4 + 25*x^2 - 2;

Triangle begins as:

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;

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

(Magma)

A084534:= func< n, k | k eq 0 select 1 else 2*(n/k)*Binomial(2*n-k-1, k-1) >;

[A084534(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 02 2022

(Sage)

def A084534(n, k): return 1 if (k==0) else 2*(n/k)*binomial(2*n-k-1, k-1)

flatten([[A084534(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 02 2022

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).