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A153229
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Weighted Fibonacci numbers.
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2
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0, 1, 0, 2, 4, 20, 100, 620, 4420, 35900, 326980, 3301820, 36614980, 442386620, 5784634180, 81393657020, 1226280710980, 19696509177020, 335990918918980, 6066382786809020, 115578717622022980, 2317323290554617020, 48773618881154822980, 1075227108896452857020
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OFFSET
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0,4
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..200
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FORMULA
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a(0) = 0, a(1) = 1, and for n>=2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1).
For n>=1, a(n) = A058006(n-1) * (-1)^(n-1).
G.f.: G(0)*x/(1+x)/2, where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
G.f.: 2*x/(1+x)/G(0), where G(k)= 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
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EXAMPLE
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a(20) = 19 * a(18) + 18 * a(19) = 19 * 335990918918980 + 18 * 6066382786809020 = 6383827459460620 + 109194890162562360 = 115578717622022980
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MAPLE
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t1 := sum(n!*x^n, n=0..100): F := series(t1/(1+x), x, 100): for i from 0 to 40 do printf(`%d, `, i!-coeff(F, x, i)) od: # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 22 2009]
# second Maple program:
a:= proc(n) a(n):= `if`(n<2, n, (n-1)*a(n-2) +(n-2)*a(n-1)) end:
seq(a(n), n=0..25); # Alois P. Heinz, May 24 2013
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MATHEMATICA
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Join[{a = 0}, Table[b = n! - a; a = b, {n, 0, 100}]] (* From Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
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PROG
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(C) unsigned long Weibonacci(unsigned int n) {
if (n == 0) return 0;
if (n == 1) return 1;
return (n - 1) * Weibonacci(n - 2) + (n - 2) * Weibonacci(n - 1); }
(PARI) a(n)=if(n, my(t=(-1)^n); -t-sum(i=1, n-1, t*=-i), 0) \\ Charles R Greathouse IV, Jun 28 2011
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CROSSREFS
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Cf. A000045, A058006
Sequence in context: A158094 A108879 A058006 * A013329 A102087 A052573
Adjacent sequences: A153226 A153227 A153228 * A153230 A153231 A153232
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KEYWORD
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nonn
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AUTHOR
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Shaojun Ying (dolphinysj(AT)gmail.com), Dec 21 2008
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EXTENSIONS
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Edited by Max Alekseyev, Jul 05 2010
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STATUS
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approved
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