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A128505
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Array for third (k=3) convolution of Chebyshev's S(n,x)=U(n,x/2) polynomials.
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2
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1, 4, 10, -4, 20, -20, 35, -60, 10, 56, -140, 60, 84, -280, 210, -20, 120, -504, 560, -140, 165, -840, 1260, -560, 35, 220, -1320, 2520, -1680, 280, 286, -1980, 4620, -4200, 1260, -56, 364, -2860, 7920, -9240, 4200, -504, 455, -4004, 12870, -18480, 11550, -2520, 84, 560, -5460, 20020, -34320
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| S3(n,x):=sum(S(n-k,x)*S2(k,x),k=0..n)= sum(a(n,m)*x^(n-2*m),m=0..floor(n/2)) with the second convolution S2(n,x) given by array A128503.
Row polynomials P3(n,x):= sum(a(n,m)*x^m,m=0..floor(n/2)) (increasing powers of x).
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LINKS
| W. Lang, First 15 rows and more.
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FORMULA
| a(n,m)= binomial(n-m+3,3)*binomial(n-m,m)*(-1)^m, m=0..floor(n/2), n>=0.
a(n,m)= binomial(m+3,3)*binomial(n-m+3,m+3)*(-1)^m, m=0..floor(n/2), n>=0.
G.f. for S3(n,x): 1/(1-x*z+z^2)^4.
G.f. for P3(n,x): 1/(1-z+x*z^2)^4.
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EXAMPLE
| [1];[4];[10,-4];[20,-20];[35,-60,10];[56,-140,60];[84,-280,210,-20];[120,-504,560,-140];...
n=4: [35,-60,10 stands also for the row polynomial P3(4,x) = 35-60*x+10*x^2.
n=4: [35,-60,10] stands also for the row polynomial P3(4,x) = 35-60*x+10*x^2.
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CROSSREFS
| Row sums (signed array) give A128506. Unsigned row sums are A001872.
Cf. A128503 (k=2 convolution).
Sequence in context: A151707 A059132 A059136 * A200454 A003564 A205016
Adjacent sequences: A128502 A128503 A128504 * A128506 A128507 A128508
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KEYWORD
| sign,tabf,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Apr 04 2007
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