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A008314 Irregular triangle read by rows: one half of the coefficients of the expansion of (2*x)^n in terms of Chebyshev T-polynomials. 5
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 (list; graph; refs; listen; history; text; internal format)
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 in Math StackExchange 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, 2. ed., Wiley, New York, 1990. p. 54-5, 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.

Math StackExchange, 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 A272172

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

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Last modified May 23 20:03 EDT 2018. Contains 304483 sequences. (Running on oeis4.)