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

0,6

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

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.

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

Example

[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)

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 *)

Crossrefs

Cf. A007318, A008311.

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

Sequence in context: A101038 A064883 A090844 * A104568 A030758 A347065

Adjacent sequences: A008311 A008312 A008313 * A008315 A008316 A008317

Keyword

nonn,tabf

Author

N. J. A. Sloane

Extensions

Name reformulated by Wolfdieter Lang, Aug 01 2014

Status

approved