A127673 -id:A127673 - cp's OEIS frontend

A122366 Triangle read by rows: T(n,k) = binomial(2*n+1,k), 0 <= k <= n.

1, 1, 3, 1, 5, 10, 1, 7, 21, 35, 1, 9, 36, 84, 126, 1, 11, 55, 165, 330, 462, 1, 13, 78, 286, 715, 1287, 1716, 1, 15, 105, 455, 1365, 3003, 5005, 6435, 1, 17, 136, 680, 2380, 6188, 12376, 19448, 24310, 1, 19, 171, 969, 3876, 11628, 27132, 50388, 75582, 92378, 1, 21

Offset: 0

Reinhard Zumkeller, Aug 30 2006

Sum of n-th row = A000302(n) = 4^n.
Central terms give A052203.
Reversal of A111418. - Philippe Deléham, Mar 22 2007
Coefficient triangle for the expansion of one half of odd powers of 2*x in terms of Chebyshev's T-polynomials: ((2*x)^(2*n+1))/2 = Sum_{k=0..n} a(n,k)*T(2*(n-k)+1,x) with Chebyshev's T-polynomials. See A053120. - Wolfdieter Lang, Mar 07 2007
The signed triangle T(n,k)*(-1)^(n-k) appears in the formula ((2*sin(phi))^(2*n+1))/2 = Sum_{k=0..n} ((-1)^(n-k))*a(n,k)*sin((2*(n-k)+1)*phi). - Wolfdieter Lang, Mar 07 2007
The signed triangle T(n,k)*(-1)^(n-k) appears therefore in the formula (4-x^2)^n = Sum_{k=0..n} ((-1)^(n-k))*a(n,k)*S(2*(n-k),x) with Chebyshev's S-polynomials. See A049310 for S(n,x). - Wolfdieter Lang, Mar 07 2007
From Wolfdieter Lang, Sep 18 2012: (Start)
The triangle T(n,k) appears also in the formula F(2*l+1)^(2*n+1) = (1/5^n)*Sum_{k=0..n} T(n,k)*F((2*(n-k)+1)*(2*l+1)), l >= 0, n >= 0, with F=A000045 (Fibonacci).
The signed triangle Ts(n,k):=T(n,k)*(-1)^k appears also in the formula
  F(2*l)^(2*n+1) = (1/5^n)*Sum_{k=0..n} Ts(n,k)*F((2(n-k)+1)*2*l), l >= 0, n >= 0, with F=A000045 (Fibonacci).
This is Lemma 2 of the K. Ozeki reference, p. 108, written for odd and even indices separately.
(End)

			.......... / 1 \ .......... =A062344(0,0)=A034868(0,0),
......... / 1 . \ ......... =T(0,0)=A034868(1,0),
........ / 1 2 . \ ........ =A062344(1,0..1)=A034868(2,0..1),
....... / 1 3 ... \ ....... =T(1,0..1)=A034868(3,0..1),
...... / 1 4 6 ... \ ...... =A062344(2,0..2)=A034868(4,0..2),
..... / 1 5 10 .... \ ..... =T(2,0..2)=A034868(5,0..2),
.... / 1 6 15 20 ... \ .... =A062344(3,0..3)=A034868(6,0..3),
... / 1 7 21 35 ..... \ ... =T(3,0..3)=A034868(7,0..3),
.. / 1 8 28 56 70 .... \ .. =A062344(4,0..4)=A034868(8,0..4),
. / 1 9 36 84 126 ..... \ . =T(4,0..4)=A034868(9,0..4).
Row n=2:[1,5,10] appears in the expansion ((2*x)^5)/2 = T(5,x)+5*T(3,x)+10*T(1,x).
Row n=2:[1,5,10] appears in the expansion ((2*cos(phi))^5)/2 = cos(5*phi)+5*cos(3*phi)+10*cos(1*phi).
The signed row n=2:[1,-5,10] appears in the expansion ((2*sin(phi))^5)/2 = sin(5*phi)-5*sin(3*phi)+10*sin(phi).
The signed row n=2:[1,-5,10] appears therefore in the expansion (4-x^2)^2 = S(4,x)-5*S(2,x)+10*S(0,x).
Triangle T(n,k) starts:
  n\k 0  1   2   3    4     5     6     7     8     9  ...
  0   1
  1   1  3
  2   1  5  10
  3   1  7  21  35
  4   1  9  36  84  126
  5   1 11  55 165  330   462
  6   1 13  78 286  715  1287  1716
  7   1 15 105 455 1365  3003  5005  6435
  8   1 17 136 680 2380  6188 12376 19448 24310
  9   1 19 171 969 3876 11628 27132 50388 75582 92378
  ...  - _Wolfdieter Lang_, Sep 18 2012
Row n=2, with F(n)=A000045(n) (Fibonacci number), l >= 0, see a comment above:
F(2*l)^5   = (1*F(10*l) - 5*F(6*l) + 10*F(2*l))/25,
F(2*l+1)^5 = (1*F(10*l+5) + 5*F(6*l+3) + 10*F(2*l+1))/25.
- _Wolfdieter Lang_, Sep 19 2012

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.

Reinhard Zumkeller, Rows n = 0..125 of triangle, 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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 795.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
K. Ozeki, On Melham's sum, The Fibonacci Quart. 46/47 (2008/2009), no. 2, 107-110.
Index entries for sequences related to Chebyshev polynomials.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A062344.

Odd numbered rows of A008314. Even numbered rows of A008314 are A127673.

a122366 n k = a122366_tabl !! n !! k
a122366_row n = a122366_tabl !! n
a122366_tabl = f 1 a007318_tabl where
   f x (_:bs:pss) = (take x bs) : f (x + 1) pss
-- Reinhard Zumkeller, Mar 14 2014

T[_, 0] = 1;
T[n_, k_] := T[n, k] = T[n-1, k-1] 2n(2n+1)/(k(2n-k+1));
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 01 2021 *)

T(n,0)=1; T(n,k) = T(n-1,k-1)*2*n*(2*n+1)/(k*(2*n-k+1)) for k > 0.
T(n,0)=1; for n > 0: T(n,1)=n+2; for n > 1: T(n,n) = T(n-1,n-2) + 3*T(n-1,n-1), T(n,k) = T(n-1,k-2) + 2*T(n-1,k-1) + T(n-1,k), 1 < k < n.
T(n,n) = A001700(n).
T(n,k) = A034868(2*n+1,k) = A007318(2*n+1,k), 0 <= k <= n;
G.f.: (2*y)/((y-1)*sqrt(1-4*x*y)-4*x*y^2+(1-4*x)*y+1). - Vladimir Kruchinin, Oct 30 2020

Chebyshev and trigonometric comments from Wolfdieter Lang, Mar 07 2007.

Typo in comments fixed, thanks to Philippe Deléham, who indicated this.

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

A273167 Numerators of coefficient triangle for expansion of x^(2n) in terms of Chebyshev polynomials of the first kind T(2m, x) (A127674).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 15, 3, 1, 35, 7, 7, 1, 1, 63, 105, 15, 45, 5, 1, 231, 99, 495, 55, 33, 3, 1, 429, 3003, 1001, 1001, 91, 91, 7, 1, 6435, 715, 1001, 273, 455, 35, 15, 1, 1, 12155, 21879, 1989, 4641, 1071, 765, 51, 153, 9, 1, 46189, 20995, 62985, 4845, 4845, 969, 4845, 285, 95, 5, 1

Offset: 0

Wolfdieter Lang, Jun 12 2016

The denominator triangle is given in A273168.
The expansion is x^(2*n) = Sum_{m=0..n} R(n, m)*Tnx(2*m, x), n >= 0, with the rational triangle R(n, m) = a(n, m)/A273168(n, m).
  Compare this with A127673.
This is equivalent to the expansion cos(x)^(2n) = Sum_{m=0..n} R(n, m)*cos(2*m*x), n >= 0. Compare this with the even numbered rows of A273496.
See A244420/A244421 for the expansion of x^(2*n+1) in terms of odd-indexed Chebyshev polynomials of the first kind.
The signed rational triangle S(n, m) = R(n, m) * (-1)^m appears in the expansion sin(x)^(2n) = Sum_{m=0..n} S(n, m) * cos(2*m*x), n >= 0. This is equivalent to the identity (1-x^2)^n =  Sum_{m=0..n} S(n, m) * T(2*m, x).

			The triangle a(n, m) begins:
n\m     0     1    2    3    4   5  6   7 8 9
0:      1
1:      1     1
2:      3     1    1
3:      5    15    3    1
4:     35     7    7    1    1
5:     63   105   15   45    5   1
6:    231    99  495   55   33   3  1
7:    429  3003 1001 1001   91  91  7   1
8:   6435   715 1001  273  455  35 15   1 1
9:  12155 21879 1989 4641 1071 765 51 153 9 1
...
The rational triangle R(n, m) begins:
n\m  0       1      2     3      4     5  ...
0:   1
1:  1/2     1/2
2:  3/8     1/2    1/8
3:  5/16   15/32   3/16  1/32
4: 35/128   7/16   7/32  1/16  1/128
5: 63/256 105/256 15/64 45/512 5/256 1/512
...
row 6: 231/1024 99/256 495/2048 55/512 33/1024 3/512 1/2048,
row 7: 429/2048 3003/8192 1001/4096 1001/8192 91/2048 91/8192 7/4096 1/8192,
row 8: 6435/32768 715/2048 1001/4096 273/2048 455/8192 35/2048 15/4096 1/2048 1/32768,
row 9: 12155/65536 21879/65536 1989/8192 4641/32768 1071/16384 765/32768 51/8192 153/131072 9/65536 1/131072,
...
n=3: x^6 = (5/16)*T(0, x) + (15/32)*T(2, x)
  +(3/16)*T(4, x) + (1/32)*T(6,x).
  cos^6(x) = (5/16) + (15/32)*cos(2*x) +
    (3/16)*cos(4*x) + (1/32)*cos(6*x).
  sin^6(x) = (5/16) - (15/32)*cos(2*x) +
    (3/16)*cos(4*x) - (1/32)*cos(6*x).

Index entries for sequences related to Chebyshev polynomials.

Cf. A127673, A127674, A244420, A244421, A273168.

T[MaxN_] := Function[{n}, With[{exp = Expand[(1/2)^(2 n) (Exp[I x] + Exp[-I x])^(2 n)]}, Prepend[ 2 Coefficient[exp, Exp[I 2 # x]] & /@ Range[1, n], exp /. {Exp[_] -> 0}]]][#] & /@ Range[0, MaxN];
T[5] // ColumnForm
T2[MaxN_] := Table[Inverse[Outer[Coefficient[#1, x, #2] &, Prepend[ChebyshevT[#, x] & /@ Range[2 MaxN], 1], Range[0, 2 MaxN]]][[n, m]], {n, 1, 2 MaxN, 2}, {m, 1, n, 2}]
T2[6] // ColumnForm (* Bradley Klee, Jun 14 2016 *)

a(n, m) = if (m == 0, numerator((1/2^(2*n-1)) * binomial(2*n,n)/2), numerator((1/2^(2*n-1))*binomial(2*n, n-m)));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(a(n,k), ", ")); print()); \\ Michel Marcus, Jun 19 2016

a(n, m) = numerator(R(n, m)), n >= 0, m = 1, ..., n, with the rationals R(n, m) given by R(n, 0) = (1/2^(2*n-1))*binomial(2*n,n)/2 and R(n ,m) = (1/2^(2*n-1))*binomial(2*n, n-m) for m =1..n, n >= 0.

A225419 Triangle read by rows: T(n,k) (0 <= k <= n) = binomial(2*n+2,k).

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, May 07 2013

Row sums are A000346.

			Triangle begins:
1,
1, 4,
1, 6, 15,
1, 8, 28, 56,
1, 10, 45, 120, 210,
1, 12, 66, 220, 495, 792,
1, 14, 91, 364, 1001, 2002, 3003,
...

Roger L. Bagula, a225419nb.txt

Cf. A127673, A062344, A034870, A000346.

Flatten[Table[Table[DifferenceRoot[Function[{y, n}, {(-(2*m + 1) + n) y[n] + n y[1 + n] == 0, y[1] == 1}]][k], {k, 1, m}], {m, 1, 10}]]
Flatten[Table[Binomial[2n+2,k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Apr 13 2014 *)

Edited by N. J. A. Sloane, May 11 2013

cp's OEIS Frontend

A122366 Triangle read by rows: T(n,k) = binomial(2*n+1,k), 0 <= k <= n.

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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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A273167 Numerators of coefficient triangle for expansion of x^(2*n) in terms of Chebyshev polynomials of the first kind T(2*m, x) (A127674).

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A225419 Triangle read by rows: T(n,k) (0 <= k <= n) = binomial(2*n+2,k).

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A273167 Numerators of coefficient triangle for expansion of x^(2n) in terms of Chebyshev polynomials of the first kind T(2m, x) (A127674).