The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A122366 Triangle read by rows: T(n,k) = binomial(2*n+1,k), 0 <= k <= n. 11
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 (list; table; graph; refs; listen; history; text; internal format)
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
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.
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
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 15 05:55 EDT 2024. Contains 373402 sequences. (Running on oeis4.)