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A127946
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Hankel transform of central coefficients of (1+k*x-3x^2)^n, k arbitrary integer.
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2
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1, -6, -108, 5832, 944784, -459165024, -669462604992, 2928229434235008, 38424226636031774976, -1512608105754026853705216, -178635992073339063368878599168, 63289660175631590117213474413627392, 67269440586795655766964092111705109663744
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OFFSET
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0,2
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COMMENTS
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Hankel transform of A098333. The Hankel transform of e.g.f. Bessel_I(0,2*sqrt(-3)x) and its k-th binomial transforms, are given by a(n). In general, the Hankel transform of e.g.f. Bessel_I(0,2*sqrt(m)x) and its binomial transforms is 2^n*m^C(n+1,2).
Let T_n denote the n X n matrix with T_n(i,j) = 3^min(i,j); then a(n) = ((-1)^floor((n+1)/2))*det(T_(n+1))/3. - Lechoslaw Ratajczak, May 16 2021
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LINKS
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FORMULA
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a(n) = (cos(Pi*n/2) - sin(Pi*n/2))*6^n*3^C(n,2) = 2^n*(-3)^C(n+1,2).
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MATHEMATICA
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Table[2^n*(-3)^Binomial[n+1, 2], {n, 0, 30}] (* G. C. Greubel, May 03 2018 *)
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PROG
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(Magma) [2^n*(-3)^Binomial(n+1, 2): n in [0..30]]; // G. C. Greubel, May 03 2018
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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