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A098331
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Expansion of 1/sqrt(1-2x+5x^2).
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6
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1, 1, -1, -5, -5, 11, 41, 29, -125, -365, -131, 1409, 3301, -155, -15625, -29485, 16115, 170035, 254525, -309775, -1813055, -2064655, 4617755, 18909175, 14903725, -61552739, -192390589, -81290561, 767919595, 1901796395, 28588201
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Central coefficients of (1+x-x^2)^n. Binomial transform of 1/sqrt(1+4x^2), or (1,0,-2,0,6,0,-20,...). Binomial transform is A098335. (-1)^nA098331(n) is the inverse binomial transform of (1,0,-2,0,6,0,-20,...).
Hankel transform is 2^n*(-1)^C(n+1,2). Hankel transform of 0,1,1,-1,-5,-5,... is F(n)*(-1)^C(n+2,2)*(2^n+0^n)/2. [From Paul Barry (pbarry(AT)wit.ie), Jan 13 2009]
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REFERENCES
| Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
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LINKS
| Eric Weisstein's World of Mathematics, Trinomial Coefficient
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FORMULA
| E.g.f. : exp(x)BesselI(0, 2*I*x), I=sqrt(-1); a(n)=sum{k=0..floor(n/2), binomial(n, 2k)binomial(2k, k)(-1)^k}; a(n)=sum{k=0..floor(n/2), binomial(n, k)binomial(n-k, k)(-1)^k); a(n)=sum{k=0..n, binomial(n, k)binomial(k, k/2)cos(pi*k/2)}.
a(0)=a(1)=1, a(n)=((2n-1)a(n-1)-5(n-1)a(n-2))/n - T. D. Noe (noe(AT)sspectra.com), Oct 19 2005
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MATHEMATICA
| a=b=1; Join[{a, b}, Table[c=((2n-1)b-5(n-1)a)/n; a=b; b=c; c, {n, 2, 30}]] (Noe)
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CROSSREFS
| Sequence in context: A121849 A164930 A173316 * A061391 A168336 A123133
Adjacent sequences: A098328 A098329 A098330 * A098332 A098333 A098334
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KEYWORD
| easy,sign
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 03 2004
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EXTENSIONS
| Corrected by T. D. Noe (noe(AT)sspectra.com), Oct 19 2005
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