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A105277
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Let F(n) denote the Fibonacci numbers, A000045: a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*F(k).
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1
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0, 1, 5, 29, 203, 1680, 16058, 173865, 2099957, 27952999, 406125305, 6389713034, 108157272720, 1958821525361, 37779732341077, 772829270394685, 16708083353842267, 380563529091632760, 9106983116342966818, 228393730451588322201, 5989333028770423686565
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OFFSET
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0,3
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COMMENTS
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If the e.g.f. of F(n) is E(x) and a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*F(k), then the e.g.f. of a(n) is E(x/(1-x))/(1-x).
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LINKS
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FORMULA
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E.g.f.: (2/sqrt(5))*exp(x/2/(1-x))*sinh(sqrt(5)*(x/2)/(1-x))/(1-x).
a(n) = (4*n - 3)*a(n-1) - 2*(n-1)*(3*n - 5)*a(n-2) + (n-2)^2*(4*n - 7)*a(n-3) - (n-3)^2*(n-2)^2*a(n-4). - Vaclav Kotesovec, Nov 13 2017
a(n) ~ n^(n + 1/4) / (sqrt(10) * phi^(1/4) * exp(n - 2*sqrt(phi*n) + phi/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 13 2017
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EXAMPLE
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F(n) = 0,1,1,2,3,5,8,13,21,34,55,...
a(3) = C(3,0)^2*3!*F(0) + C(3,1)^2*2!*F(1) + C(3,2)^2*1!*F(2) + C(3,3)^2*0!*F(3) = 1*6*0 + 9*2*1 + 9*1*1 + 1*1*2 = 0 + 18 + 9 + 2 = 29.
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MAPLE
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b[0]:=0:b[1]:=1:for n from 2 to 30 do b[n]:=b[n-1]+b[n-2] od:
seq(sum('binomial(n, k)^2*(n-k)!*b[k]', 'k'=0..n), n=0..30);
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MATHEMATICA
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Table[Sum[Binomial[n, k]^2 * (n-k)! * Fibonacci[k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 13 2017 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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