A008311 -id:A008311 - cp's OEIS frontend

A100257 Triangle of expansions of 2^(k-1)*x^k in terms of T(n,x), in descending degrees n of T, with T the Chebyshev polynomials.

1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 4, 0, 3, 1, 0, 5, 0, 10, 0, 1, 0, 6, 0, 15, 0, 10, 1, 0, 7, 0, 21, 0, 35, 0, 1, 0, 8, 0, 28, 0, 56, 0, 35, 1, 0, 9, 0, 36, 0, 84, 0, 126, 0, 1, 0, 10, 0, 45, 0, 120, 0, 210, 0, 126, 1, 0, 11, 0, 55, 0, 165, 0, 330, 0, 462, 0, 1, 0, 12, 0, 66, 0, 220, 0

Offset: 0

Ralf Stephan, Nov 13 2004

			x^0 = T(0,x)
x^1 = T(1,x) + 0T(0,x)
2x^2 = T(2,x) + 0T(1,x) + 1T(0,x)
4x^3 = T(3,x) + 0T(2,x) + 3T(1,x) + 0T(0,x)
8x^4 = T(4,x) + 0T(3,x) + 4T(2,x) + 0T(1,x) + 3T(0,x)
16x^5 = T(5,x) + 0T(4,x) + 5T(3,x) + 0T(2,x) + 10T(1,x) + 0T(0,x)

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.

Vincenzo Librandi, Table of n, a(n) for n = 0..6104
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].
H. J. Brothers, Pascal's Prism: Supplementary Material.
Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, see page 175.
Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 97.
Index entries for sequences related to Chebyshev polynomials.

Without zeros: A008311. Row sums are A011782. Cf. A092392.

Diagonals are (with interleaved zeros) twice A001700, A001791, A002054, A002694, A003516, A002696, A030053, A004310, A030054, A004311, A030055, A004312, A030056, A004313.

a[k_, n_] := If[k == 1, 1, If[EvenQ[n] || k < 0 || n > k, 0, If[n >= k - 1, Binomial[2*Floor[k/2], Floor[k/2]]/2, Binomial[k - 1, Floor[n/2]]]]];
Table[a[k, n], {k, 1, 13}, {n, 1, k}] // Flatten (* Jean-François Alcover, May 04 2017, translated from PARI *)

a(k,n)=if(k==1,1,if(n%2==0||k<0||n>k,0,if(n>=k-1,binomial(2*floor(k/2),floor(k/2))/2,binomial(k-1,floor(n/2)))))

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

Original entry on oeis.org

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

N. J. A. Sloane

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

			[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)

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.

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.

Cf. A007318, A008311.

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

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

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 *)

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

Name reformulated by Wolfdieter Lang, Aug 01 2014

A127673 One half of even powers of 2*x in terms of Chebyshev's T-polynomials.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 6, 15, 10, 1, 8, 28, 56, 35, 1, 10, 45, 120, 210, 126, 1, 12, 66, 220, 495, 792, 462, 1, 14, 91, 364, 1001, 2002, 3003, 1716, 1, 16, 120, 560, 1820, 4368, 8008, 11440, 6435, 1, 18, 153, 816, 3060, 8568, 18564, 31824, 43758, 24310, 1, 20, 190

Offset: 0

Wolfdieter Lang, Mar 07 2007

See A122366 for one half of odd powers of 2*x in terms of Chebyshev's T-polynomials.
This is, for n >= 1, the left half of Pascal's triangle for even rows with the central coefficients divided by 2.
The signed version of this triangle, b(n,k) := a(n,k)*(-1)^(n-k), appears in the formula (1/2)*(2*sin(phi))^(2*n) = (Sum_{k=0..n-1} b(n,k)*cos(2*(n-k)*phi)) + a(n,n).
Correspondingly, (1/2)*(4*(1-x^2))^n = (Sum_{k=0..n-1} b(n,k)*T(2*(n-k),x)) + a(n,n).
The proofs follow from Euler's formula 2*x = 2*cos(phi) = exp(i*phi) + exp(-i*phi) or 2*sqrt(1-x^2) = 2*sin(phi) = (exp(i*phi) - exp(-i*phi))/i and the binomial formula.

			[1/2];
[ 1, 1];
[ 1, 4,  3];
[ 1, 6, 15, 10];
[ 1, 8, 28, 56, 35];
...
Row n=3: [1, 6, 15, 20/2 = 10] appears in ((2*x)^6)/2 = 1*T(6,x) + 6*T(4,x) + 15*T(2,x) + 10.
Row n=3: [1, 6, 15, 20/2 = 10] appears in ((2*cos(phi))^6)/2 = 1*cos(6*phi) + 6*cos(4*phi) + 15*cos(2*phi) + 10.
The signed row n=3, [-1, 6, -15, +20/2 = 10], appears in ((4*(1-x^2))^3)/2 = -1*T(6,x) + 6*T(4,x) - 15*T(2,x) + 10).
The signed row n=3, [-1, 6, -15, +20/2 = 10], appears in ((2*sin(phi))^6)/2 = -1*cos(6*phi) + 6*cos(4*phi) - 15*cos(2*phi) + 10.

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

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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 795.
Index entries for sequences related to Chebyshev polynomials.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A122366, A008314, A008311.

a(n,k) = binomial(2*n,k), k=0..n-1 and a(n,n) = binomial(2*n,n)/2, n >= 1. Instead of a(0,0)=1 one should take 1/2.

A380113 Triangle read by rows: The inverse matrix of the central factorials A370707, row n normalized by (-1)^(n - k)*A370707(n, n).

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 10, 15, 6, 1, 35, 56, 28, 8, 1, 126, 210, 120, 45, 10, 1, 462, 792, 495, 220, 66, 12, 1, 1716, 3003, 2002, 1001, 364, 91, 14, 1, 6435, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 24310, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1

Offset: 0

Peter Luschny, Jan 12 2025

The inverse matrix of A370707 is a rational matrix and the normalization serves to make it a matrix over the integers. Note that the normalization factor A370707(n, n) = FallingFactorial(n, n) * RisingFactorial(n, n) extends A002674 to n = 0.

			Triangle starts:
  [0] [    1]
  [1] [    1,     1]
  [2] [    3,     4,     1]
  [3] [   10,    15,     6,     1]
  [4] [   35,    56,    28,     8,    1]
  [5] [  126,   210,   120,    45,   10,    1]
  [6] [  462,   792,   495,   220,   66,   12,   1]
  [7] [ 1716,  3003,  2002,  1001,  364,   91,  14,   1]
  [8] [ 6435, 11440,  8008,  4368, 1820,  560, 120,  16,  1]
  [9] [24310, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1]
.
Row 3 of the matrix inverse of the central factorials is [-1/36, 1/24, -1/60, 1/360]. Normalized with (-1)^(n-k)*360 gives row 3 of T.

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Variant: A094527.

Cf. A370707, A002674, A008311, A088218 and A110556 (column 0), A081294 (row sums), A000007 (alternating row sums), A005810 (central terms).

T := (n, k) -> if n = k then 1 elif k = 0 then binomial(2*n, n - k)/2 else binomial(2*n, n - k) fi: seq(seq(T(n, k), k = 0..n), n = 0..9);

A380113[n_, k_] := Binomial[2*n, n - k]/(Boole[k == 0 && n > 0] + 1);
Table[A380113[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 13 2025 *)

def Trow(n):
    def cf(n, k): return falling_factorial(n, k)*rising_factorial(n, k)
    def w(n): return factorial(n)*rising_factorial(n, n)
    m = matrix(QQ, n + 1, lambda x, y: cf(x, y)).inverse()
    return [(-1)^(n-k)*w(n)*m[n, k] for k in range(n+1)]
for n in range(10): print(Trow(n))

T(n, k) = (-1)^(n - k) * ff(n, n) * rf(n, n) * M^(-1)(ff(n, k) * rf(n, k)) where ff denotes the falling factorial, rf the rising factorial and M^(-1)(t(n, k)) the matrix inverse to the matrix with entries t(n, k).
T(n, k) = binomial(2*n, n - k) for 0 < k < n. T(n, n) = 1; T(n, 0) = (-1)^n*binomial(-n, n).
Sum_{k=0..n} T(n, k)*cos(k*x) = 2^(n-1)*(cos(x)+1)^n. (After Philippe Deléham in A008311).

cp's OEIS Frontend

A111417 a(n) = A034869(n) - A008311(n).

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A100257 Triangle of expansions of 2^(k-1)*x^k in terms of T(n,x), in descending degrees n of T, with T the Chebyshev polynomials.

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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.

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A127673 One half of even powers of 2*x in terms of Chebyshev's T-polynomials.

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A380113 Triangle read by rows: The inverse matrix of the central factorials A370707, row n normalized by (-1)^(n - k)*A370707(n, n).

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