0,3

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,k) = A034868(2*n+1,k) = A007318(2*n+1,k), 0<=k<=n;

sum of n-th row = A000302(n) = n^4;

central terms give A052203; T(n,n) = A001700(n).

Reversal of A111418. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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(a(n,k)*T(2*(n-k)+1,x),k=0..n) with Chebyshev's T-polynomials. See A053120. - W. Lang, Mar 07 2007.

The signed triangle a(n,k)*(-1)^(n-k) appears in the formula (2*sin(phi))^(2*n+1))/2 = sum(((-1)^(n-k))*a(n,k)*sin((2*(n-k)+1)*phi),k=0..n) - W. Lang, Mar 07 2007.

The signed triangle a(n,k)*(-1)^(n-k) appears therefore in the formula (4-x^2)^n = sum(((-1)^(n-k))*a(n,k)*S(2*(n-k),x),k=0..n) with Chebyshev's S-polynomials. See A049310 for S(n,x). - W. Lang, Mar 07 2007.

T. J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. p. 54-5, 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].

Index entries for triangles and arrays related to Pascal's triangle

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.

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.

.......... / 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).

Cf. A062344.

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

Sequence in context: A055199 A146916 A146255 this_sequence A103327 A065229 A093905

Adjacent sequences: A122363 A122364 A122365 this_sequence A122367 A122368 A122369

nonn,tabl

Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 30 2006

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