A008314 Irregular triangle read by rows: one half of the coefficients of the expansion of (2*x)^n in terms of Chebyshev T-polynomials.

1, 1, 1, 1, 1, 3, 1, 4, 3, 1, 5, 10, 1, 6, 15, 10, 1, 7, 21, 35, 1, 8, 28, 56, 35, 1, 9, 36, 84, 126, 1, 10, 45, 120, 210, 126, 1, 11, 55, 165, 330, 462, 1, 12, 66, 220, 495, 792, 462, 1, 13, 78, 286, 715, 1287, 1716, 1, 14, 91, 364, 1001, 2002, 3003, 1716, 1, 15, 105, 455, 1365, 3003, 5005

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabf

Comments

The entry a(0,0) should actually be 1/2.

The row lengths of this array are [1,1,2,2,3,3,...] = A004526.

Row k also counts the binary strings of length k that have 0, 2 up to 2*floor(k/2) 'unmatched symbols'. See contributions by Marc van Leeuwen at the Mathematics Stack Exchange link. - Wouter Meeussen, Apr 17 2013

For n >= 1, T(n,k) is the coefficient of cos((n-2k)x) in the expression for 2^(n-1)*cos(x)^n as a sum of cosines of multiples of x. It is binomial(n,k) if k < n/2, while T(n,n/2) = binomial(n,n/2)/2 if n is even. - Robert Israel, Jul 25 2016

Examples

			[1/2], [1], [1,2/2=1], [1,3], [1,4,6/2=3], [1,5,10], [1,6,15,20/2=10],...
From _Wolfdieter Lang_, Aug 01 2014: (Start)
This irregular triangle begins (even n has falling even T-polynomial indices, odd n has falling odd T-indices):
n\k  1  2   3   4     5     6     7     8 ...
0: 1/2 (but a(0,1) = 1)
1:   1
2:   1  1
3:   1  3
4:   1  4   3
5:   1  5  10
6:   1  6  15  10
7:   1  7  21  35
8:   1  8  28  56    35
9:   1  9  36  84   126
10:  1 10  45 120   210   126
11:  1 11  55 165   330   462
12:  1 12  66 220   495   792   462
13:  1 13  78 286   715  1287  1716
14:  1 14  91 364  1001  2002  3003  1716
15:  1 15 105 455  1365  3003  5005  6435
...
(2*x)^5 = 2*(1*T_5(x) + 5*T_3(x) + 10*T_1(x)),
(2*x)^6 = 2*(1*T_6(x) + 6*T_4(x) + 15*T_3(x) + 10*T_0(x)).
(End)
		

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.
• T. J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2nd ed., Wiley, New York, 1990, pp. 54-55, Ex. 1.5.31.

Links

• Robert Israel, Table of n, a(n) for n = 0..10099 (rows 0 to 199, 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].
• Suyoung Choi and Hanchul Park, A new graph invariant arises in toric topology, arXiv preprint arXiv:1210.3776 [math.AT], 2012.
• Mathematics Stack Exchange, Bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n, April-March 2013.
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A007318, A008311.

Bisection triangles: A122366 (odd numbered rows), A127673 (even numbered rows).

Programs

Maple

F:= proc(n) local q;
  q:= combine(2^(n-1)*cos(t)^n,trig);
  if n::even then
     seq(coeff(q,cos((n-2*j)*t)),j=0..n/2-1),eval(q,cos=0)
  else
     seq(coeff(q,cos((n-2*j)*t)),j=0..(n-1)/2)
  fi
end proc:
1, seq(F(n),n=1..15); # Robert Israel, Jul 25 2016

Mathematica

Table[(c/@ Range[n,0,-2]) /. Flatten[Solve[Thread[CoefficientList[Expand[1/2*(2*x)^n -Sum[c[k] ChebyshevT[k,x],{k,0,n}]],x]==0]]],{n,16}];
(* or with combinatorics *)
match[li:{(1|-1)..}]:= Block[{it=li,rot=0}, While[Length[Union[Join[it,{"(",")"}]]]>3, rot++; it=RotateRight[it //.{a___,1,b___String,-1,c___} ->{a,"(",b,")",c}]]; RotateLeft[it,rot] /. {(1|-1)->0, "("->1,")"->-1}];
Table[Last/@ Sort@ Tally[Table[Tr[Abs@ match[-1+2*IntegerDigits[n,2]]], {n,2^(k-1), 2^k-1}]], {k,1,16}]; (* Wouter Meeussen, Apr 17 2013 *)

Formula

a(n,k) are the M_3 multinomial numbers A036040 for the partitions with m = 1 and 2 parts (in Abramowitz-Stegun order). - Wolfdieter Lang, Aug 01 2014

Extensions

Name reformulated by Wolfdieter Lang, Aug 01 2014