A128494 -id:A128494 - cp's OEIS frontend

A128498 Fourth column (m=3) of triangle A128494.

1, 1, -3, -3, 7, 7, -13, -13, 22, 22, -34, -34, 50, 50, -70, -70, 95, 95, -125, -125, 161, 161, -203, -203, 252, 252, -308, -308, 372, 372, -444, -444, 525, 525, -615, -615, 715, 715, -825, -825, 946, 946, -1078, -1078, 1222, 1222, -1378, -1378, 1547, 1547, -1729, -1729, 1925, 1925, -2135, -2135

Offset: 0

Wolfdieter Lang, Apr 04 2007

Unsigned, this is the repeated sequence A002623.

Index entries for linear recurrences with constant coefficients, signature (1,-4,4,-6,6,-4,4,-1,1).

Cf. A008642 (unsigned column m=2). A128499 (column m=4).

CoefficientList[Series[1/((1-x)(1+x^2)^4),{x,0,60}],x] (* or *) LinearRecurrence[{1,-4,4,-6,6,-4,4,-1,1},{1,1,-3,-3,7,7,-13,-13,22},60] (* Harvey P. Dale, Jul 04 2021 *)

Vec(1/((1-x)*(1+x^2)^4) + O(x^50)) \\ Michel Marcus, Mar 16 2015

G.f.: 1/((1-x)*(1+x^2)^4).
a(2*k) = a(2*k+1)= ((-1)^k)*A002623(n), k>=0.
a(n) = (-1)^((2*n-1+(-1)^n)/4)*((n+2)*(n+7)*(2*n+9)+3*(n+3)*(n+6)*(-1)^n+12*(-1)^((2*n-1+(-1)^n)/4))/192. - Luce ETIENNE, Mar 13 2015

A128499 Fifth column (m=4) of triangle A128494.

Original entry on oeis.org

1, 1, -4, -4, 11, 11, -24, -24, 46, 46, -80, -80, 130, 130, -200, -200, 295, 295, -420, -420, 581, 581, -784, -784, 1036, 1036, -1344, -1344, 1716, 1716, -2160, -2160, 2685, 2685, -3300, -3300, 4015, 4015, -4840, -4840, 5786, 5786, -6864, -6864, 8086, 8086, -9464, -9464, 11011, 11011, -12740

Offset: 0

Wolfdieter Lang, Apr 04 2007

Unsigned, this is the repeated sequence A001752.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-5,5,-10,10,-10,10,-5,5,-1,1).

Cf. A128498 (column m=3).

LinearRecurrence[{1,-5,5,-10,10,-10,10,-5,5,-1,1},{1,1,-4,-4,11,11,-24,-24,46,46,-80},60] (* Harvey P. Dale, Aug 26 2023 *)

Vec(-1/((x-1)*(x^2+1)^5) + O(x^100)) \\ Colin Barker, Mar 14 2015

G.f.: -1 / ((x-1)*(x^2+1)^5). - Corrected by Colin Barker, Mar 14 2015
a(2*k) = a(2*k+1) = ((-1)^k)*A001752(n), k>=0.
a(n) = ((2*n^4+44*n^3+334*n^2+1012*n+993)*(-1)^((2*n-1+(-1)^n)/4)+(4*n^3+66*n^2+332*n+495)*(-1)^((6*n-1+(-1)^n)/4)+48)/1536. - Luce ETIENNE, Mar 14 2015

A128495 Coefficient table for sums of squares of Chebyshev's S-Polynomials.

Original entry on oeis.org

1, 1, 1, 2, -1, 1, 2, 3, -3, 1, 3, -3, 8, -5, 1, 3, 6, -16, 17, -7, 1, 4, -6, 30, -45, 30, -9, 1, 4, 10, -50, 103, -98, 47, -11, 1, 5, -10, 80, -211, 269, -183, 68, -13, 1, 5, 15, -120, 399, -651, 588, -308, 93, -15, 1, 6, -15, 175, -707, 1432, -1644, 1136, -481, 122, -17, 1, 6, 21, -245, 1190, -2920, 4132, -3608

Offset: 0

Wolfdieter Lang Apr 04 2007

See A049310 for the coefficient table of Chebyshev's S(n,x)=U(n,x/2) polynomials.

The triangle for the coefficients of x^2 in S(n,x)^2 is A158454. - Wolfdieter Lang, Oct 18 2012

			[1]; [1,1]; [2,-1,1]; [2,3,-3,1]; [3,-3,8,-5,1]; [3,6,-16,17,-7,1]; ...
Row polynomial S(2;4,x)=3-3*x^2+8*x^4-5*x^6+x^8 = sum(S(k,x)^2,k=0..4).
(4+2-T(4+1,x/2)*U(4+1,x/2))/(2*(1-(x/2)^2))= S(2;4,x)

W. Lang, First 15 rows.

Row sums (signed) look like: A004523. Row sums (unsigned): A128496.

Cf. A128494 =S(1; n, m).

S(2;n,x):=sum(S(k,x)^2,k=0..n)=sum(a(n,m)*x^(2*m),m=0..n), n>=0.

a(n,m)=[x^m](n+2-T(n+1,x/2)*U(n+1,x/2))/(2*(1-(x/2)^2)).

cp's OEIS Frontend

A128498 Fourth column (m=3) of triangle A128494.

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A128499 Fifth column (m=4) of triangle A128494.

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A128495 Coefficient table for sums of squares of Chebyshev's S-Polynomials.

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