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

0,3

Comments

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)

References

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

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

Formula

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

Example

.......... / 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

Mathematica

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 *)

Prog

(Haskell)
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

Crossrefs

Cf. A062344.

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

Sequence in context: A146916 A146255 A331432 * A228781 A103327 A177463

Adjacent sequences: A122363 A122364 A122365 * A122367 A122368 A122369

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Aug 30 2006

Extensions

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

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

Status

approved