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A181878
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Coefficient array for square of Chebyshev S-polynomials.
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5
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1, 1, 1, -2, 1, 4, -4, 1, 1, -6, 11, -6, 1, 9, -24, 22, -8, 1, 1, -12, 46, -62, 37, -10, 1, 16, -80, 148, -128, 56, -12, 1, 1, -20, 130, -314, 367, -230, 79, -14, 1, 25, -200, 610, -920, 771, -376, 106, -16, 1, 1, -30, 295, -1106, 2083, -2232, 1444, -574, 137, -18, 1, 36, -420, 1897, -4352, 5776, -4744, 2486, -832, 172, -20, 1, 1, -42, 581, -3108, 8518, -13672, 13820, -9142, 4013, -1158, 211, -22, 1
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OFFSET
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0,4
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COMMENTS
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For the coefficients of Chebyshev polynomials S(n,x) see A049310.
The row length sequence for this array is A109613=[1,1,3,3,5,5,...].
The row polynomials (in x^2) for even row numbers are
S(2*k,x)^2 = sum(a(2*k,m)*x^(2*m),m=0..2*k), k>=0.
For odd row numbers the row polynomials (in x^2) are
(S(2*k+1,x)^2)/x^2 = sum(a(2*k+1,m)*x^(2*m),m=0..2*k), k>=0.
The o.g.f. for the polynomials S(n,x)^2 is
S(x,z):=((1+z)/(1-z))/(1 + (2-x^2)z +z^2). See the link for a proof. Therefore the coefficients constitute the Riordan array (1/(1-x^2),x/(1+x)^2) found as A158454.
The o.g.f. for (S(2*k,sqrt(x))^2 is
(1-2(1-x)z+z^2)/((1-z)*(1 - (2-4x+x^2)z + z^2)).
The o.g.f. for ((S(2*k+1,sqrt(x))^2)/x is
((1+z)/(1-z))/(1 - (2-4x+x^2)z + z^2).
The row sums A011655(n+1) are the same as those for the triangle A158454.
The alternating row sums for even numbered rows ((-1)^n)*A007598(n+1) coincide with those of triangle A158454. For odd row numbers n=2k+1 these sums are A049684(k+1), k>=0 (squares of even indexed Fibonacci numbers).
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LINKS
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Table of n, a(n) for n=0..84.
Wolfdieter Lang, First ten rows with more details and proofs.
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FORMULA
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a(2*k,m) = ((-1)^m)*sum(binomial(2*k+m-1-2*j,2*m-1),j=0..k), k>=0.
a(2*k+1,m) = ((-1)^m)*sum(binomial(2*k+1+m-2*j,2*m+1),j=0..k), k>=0.
This derives from the formula for the entries of the Riordan array A158454.
For the o.g.f.s see the comment.
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EXAMPLE
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The irregular triangle a(n,m) begins:
n\m 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 1
2: 1 -2 1
3: 4 -4 1
4: 1 -6 11 -6 1
5: 9 -24 22 -8 1
6: 1 -12 46 -62 37 -10 1
7: 16 -80 148 -128 56 -12 1
8: 1 -20 130 -314 367 -230 79 -14 1
9: 25 -200 610 -920 771 -376 106 -16 1
10: 1 -30 295 -1106 2083 -2232 1444 -574 137 -18 1
... Reformatted and extended by Wolfdieter Lang, Nov 24 2012
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CROSSREFS
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Cf. A158454, A129818.
Sequence in context: A004175 A136756 A214670 * A137593 A009949 A070785
Adjacent sequences: A181875 A181876 A181877 * A181879 A181880 A181881
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KEYWORD
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sign,easy,tabf
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AUTHOR
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Wolfdieter Lang, Dec 22 2010
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EXTENSIONS
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Corrected by Wolfdieter Lang, Jan 21 2011.
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STATUS
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approved
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