A100622 Expansion of e.g.f. exp( (1+2*x-sqrt(1-4*x))/4).
1, 1, 2, 10, 94, 1286, 22876, 499612, 12925340, 386356924, 13099953016, 496719289496, 20825694943912, 956599393819720, 47772070664027984, 2577034852683364816, 149335440671982405136, 9251650217381166689552, 610194993478502245703200, 42688019374465782644235424
Offset: 0
Keywords
Examples
G.f. = 1 + x + 2*x^2 + 10*x^3 + 94*x^4 + 1286*x^5 + 22876*x^6 + 499612*x^7 + ...
Links
- Lee A. Newberg, Table of n, a(n) for n = 0..100
- Lee Aaron Newberg, Finding, Evaluating and Counting DNA Physical Maps, Ph.D. Thesis, University of California, 1993, Berkeley, CA.
- Lee A. Newberg, The Number of Clone Orderings, Discrete Applied Mathematics, Vol. 69 (1996) pp. 233-245.
Programs
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Maple
a := proc(n) option remember: if n = 0 then factorial(0) elif n = 1 then factorial(1) elif n = 2 then factorial(2) elif n >= 3 then (4*n-5)*procname(n-1) - (4*n-7)*procname(n-2) + (n-2)*procname(n-3) fi; end: seq(a(n), n = 0..250); # Muniru A Asiru, Jan 23 2018
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Mathematica
CoefficientList[Series[Exp[(1+2*x-Sqrt[1-4*x])/4], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *) -
PARI
{a(n) = if( n<0, 0, n! * polcoeff( exp( (1 + 2*x - sqrt(1 - 4*x + x * O(x^n))) / 4), n))}; /* Michael Somos, Jan 03 2015 */
Formula
a(n) = n! for n = 0, 1, 2. a(n) = (4n-5) * a(n-1) - (4n-7) * a(n-2) + (n-2) * a(n-3) for n > 2. - Lee A. Newberg, Oct 18 2006
a(n) ~ n^(n-1)*exp(3/8-n)*2^(2*n-5/2). - Vaclav Kotesovec, Jun 26 2013
Given e.g.f. A(x), then A'(x) / A(x) = g.f. for A088218. - Michael Somos, Jan 03 2015
E.g.f.: exp( (1 + 2*x - sqrt(1 - 4*x)) / 4). - Michael Somos, Jan 03 2015
0 = a(n)*(+a(n+1) - 5*a(n+2) + 7*a(n+3) - a(n+4)) + a(n+1)*(-3*a(n+1) + 17*a(n+2) - 23*a(n+3) + 4*a(n+4)) + a(n+2)*(-8*a(n+2) + 12*a(n+3) - 4*a(n+4)) + a(n+3)*(+4*a(n+3)) for all n>-2. - Michael Somos, Jan 03 2015
Comments