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A091520
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Expansion of 1 / ((1 - 4*x) * sqrt(1 + 4*x)) in powers of x.
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1
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1, 2, 14, 36, 214, 604, 3340, 9928, 52582, 161708, 831588, 2620920, 13187836, 42350744, 209519576, 682960784, 3332923526, 10998087884, 53067486836, 176924683544, 845545262996, 2843923177544, 13479791673896, 45685735967984
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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FORMULA
| G.f.: 1 / ((1 - 4*x) * sqrt(1 + 4*x)). n * a(n) = 2 * a(n-1) + 8 * (2*n - 1) * a(n-2).
a(n) = 4^n * (Sum_{k=0, n} binomial( 2*k, k) / (-4)^k).
a(n) = A003148(n) * 2^n / n!. - Michael Somos Mar 17 2011
Asymptotics: a(n) ~ 4^n / sqrt(2).
G.f. y = A(x) satisfies 0 = (16*x^2 - 1) * y' + (24*x + 2) * y and 0 = y'^3 + 8 * y'^2 * y^3 + 216 * y^5 - 256 * y^7.
G.f.: 1/((1-4*x)*sqrt(1+4*x)) = 1/(1-4*x+2*x*(1-4*x)/G(0)) ; G(k) = 1 + x/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 21 2011
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EXAMPLE
| 1 + 2*x + 14*x^2 + 36*x^3 + 214*x^4 + 604*x^5 + 3340*x^6 + 9928*x^7 + ...
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PROG
| (PARI) {a(n) = if( n<0, 0, 4^n * sum( k=0, n, binomial(2*k, k) / (-4)^k))}
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CROSSREFS
| Sequence in context: A134647 A004117 A135706 * A108597 A062548 A067293
Adjacent sequences: A091517 A091518 A091519 * A091521 A091522 A091523
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Jan 18 2004
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