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

%I A122366 #64 Jul 29 2025 08:37:24
%S A122366 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,
%T A122366 715,1287,1716,1,15,105,455,1365,3003,5005,6435,1,17,136,680,2380,
%U A122366 6188,12376,19448,24310,1,19,171,969,3876,11628,27132,50388,75582,92378,1,21
%N A122366 Triangle read by rows: T(n,k) = binomial(2*n+1,k), 0 <= k <= n.
%C A122366 Sum of n-th row = A000302(n) = 4^n.
%C A122366 Central terms give A052203.
%C A122366 Reversal of A111418. - _Philippe Deléham_, Mar 22 2007
%C A122366 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
%C A122366 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
%C A122366 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
%C A122366 From _Wolfdieter Lang_, Sep 18 2012: (Start)
%C A122366 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).
%C A122366 The signed triangle Ts(n,k):=T(n,k)*(-1)^k appears also in the formula
%C A122366   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).
%C A122366 This is Lemma 2 of the K. Ozeki reference, p. 108, written for odd and even indices separately.
%C A122366 (End)
%D A122366 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.
%H A122366 Reinhard Zumkeller, <a href="/A122366/b122366.txt">Rows n = 0..125 of triangle, flattened</a>
%H A122366 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A122366 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 795.
%H A122366 C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages)
%H A122366 K. Ozeki, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/46_47-2/Ozeki.pdf">On Melham's sum</a>, The Fibonacci Quart. 46/47 (2008/2009), no. 2, 107-110.
%H A122366 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A122366 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A122366 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.
%F A122366 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.
%F A122366 T(n,n) = A001700(n).
%F A122366 T(n,k) = A034868(2*n+1,k) = A007318(2*n+1,k), 0 <= k <= n;
%F A122366 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
%e A122366 .......... / 1 \ .......... =A062344(0,0)=A034868(0,0),
%e A122366 ......... / 1 . \ ......... =T(0,0)=A034868(1,0),
%e A122366 ........ / 1 2 . \ ........ =A062344(1,0..1)=A034868(2,0..1),
%e A122366 ....... / 1 3 ... \ ....... =T(1,0..1)=A034868(3,0..1),
%e A122366 ...... / 1 4 6 ... \ ...... =A062344(2,0..2)=A034868(4,0..2),
%e A122366 ..... / 1 5 10 .... \ ..... =T(2,0..2)=A034868(5,0..2),
%e A122366 .... / 1 6 15 20 ... \ .... =A062344(3,0..3)=A034868(6,0..3),
%e A122366 ... / 1 7 21 35 ..... \ ... =T(3,0..3)=A034868(7,0..3),
%e A122366 .. / 1 8 28 56 70 .... \ .. =A062344(4,0..4)=A034868(8,0..4),
%e A122366 . / 1 9 36 84 126 ..... \ . =T(4,0..4)=A034868(9,0..4).
%e A122366 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).
%e A122366 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).
%e A122366 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).
%e A122366 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).
%e A122366 Triangle T(n,k) starts:
%e A122366   n\k 0  1   2   3    4     5     6     7     8     9  ...
%e A122366   0   1
%e A122366   1   1  3
%e A122366   2   1  5  10
%e A122366   3   1  7  21  35
%e A122366   4   1  9  36  84  126
%e A122366   5   1 11  55 165  330   462
%e A122366   6   1 13  78 286  715  1287  1716
%e A122366   7   1 15 105 455 1365  3003  5005  6435
%e A122366   8   1 17 136 680 2380  6188 12376 19448 24310
%e A122366   9   1 19 171 969 3876 11628 27132 50388 75582 92378
%e A122366   ...  - _Wolfdieter Lang_, Sep 18 2012
%e A122366 Row n=2, with F(n)=A000045(n) (Fibonacci number), l >= 0, see a comment above:
%e A122366 F(2*l)^5   = (1*F(10*l) - 5*F(6*l) + 10*F(2*l))/25,
%e A122366 F(2*l+1)^5 = (1*F(10*l+5) + 5*F(6*l+3) + 10*F(2*l+1))/25.
%e A122366 - _Wolfdieter Lang_, Sep 19 2012
%t A122366 T[_, 0] = 1;
%t A122366 T[n_, k_] := T[n, k] = T[n-1, k-1] 2n(2n+1)/(k(2n-k+1));
%t A122366 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 01 2021 *)
%o A122366 (Haskell)
%o A122366 a122366 n k = a122366_tabl !! n !! k
%o A122366 a122366_row n = a122366_tabl !! n
%o A122366 a122366_tabl = f 1 a007318_tabl where
%o A122366    f x (_:bs:pss) = (take x bs) : f (x + 1) pss
%o A122366 -- _Reinhard Zumkeller_, Mar 14 2014
%Y A122366 Cf. A062344.
%Y A122366 Odd numbered rows of A008314. Even numbered rows of A008314 are A127673.
%K A122366 nonn,tabl
%O A122366 0,3
%A A122366 _Reinhard Zumkeller_, Aug 30 2006
%E A122366 Chebyshev and trigonometric comments from _Wolfdieter Lang_, Mar 07 2007.
%E A122366 Typo in comments fixed, thanks to _Philippe Deléham_, who indicated this.