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A232546
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Expansion of (1 - 12*x)^(3/2) in powers of x.
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2
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1, -18, 54, 108, 486, 2916, 20412, 157464, 1299078, 11258676, 101328084, 939587688, 8926083036, 86514343272, 852784240824, 8527842408240, 86344404383430, 883760374277460, 9132190534200420, 95167038198509640, 999253901084351220, 10563541240034570040
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OFFSET
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0,2
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COMMENTS
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By analytic continuation to the entire complex plane there exist regularized values for divergent sums such as:
Sum_{k>=0} a(k)^2/16^k = 2F1(-3/2,-3/2,1,9).
Sum_{k>=0} a(k) / 6^k = -i. (End)
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LINKS
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FORMULA
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0 = a(n+2)*(a(n+1) - 42*a(n)) + 18*a(n+1)*(a(n+1) + 8*a(n)) for all n in Z.
For n>=2, a(n) = 4*3^(n+1)*(2*n-4)! / ((n-2)!*n!). - Vaclav Kotesovec, Apr 02 2017
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EXAMPLE
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G.f. = 1 - 18*x + 54*x^2 + 108*x^3 + 486*x^4 + 2916*x^5 + 20412*x^6 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (1 - 12 x)^(3/2), {x, 0, n}];
Table[9/Sqrt[Pi] 12^n Gamma[-1/2 + n]/Gamma[2 + n], {n, -1, 20}] (* Ralf Steiner, Apr 01 2017 *)
Flatten[{1, -18, Table[4*3^(n+1)*(2*n-4)!/((n-2)!*n!), {n, 2, 25}]}] (* Vaclav Kotesovec, Apr 02 2017 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( (1 - 12 * x + x * O(x^n))^(3/2), n))};
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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