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A125187
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Number of Dumont permutations of the first kind of length 2n avoiding the patterns 1423 and 4132.
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3
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1, 1, 3, 12, 52, 232, 1049, 4777, 21845, 100159, 460023, 2115350, 9735205, 44829766, 206526972, 951759621, 4387156587, 20226421380, 93264500832, 430091815527, 1983549213861, 9148582037193, 42197572190160, 194643215702835
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OFFSET
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0,3
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COMMENTS
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[1, 3, 12, 52, 232, ...] is INVERT transform of [1, 2, 27, 108, 440, ...] A026726. - Michael Somos, Apr 15 2012
HANKEL transform of sequence and the sequence omitting a(0) is the odd and even bisections of Fibonacci numbers respectively. This is the unique sequence with that property. - Michael Somos, Apr 15 2012
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LINKS
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FORMULA
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G.f.: [2-(1+x)C(x)]/[2-x-(1+x)C(x)], where C(x)=(1-sqrt(1-4x))/(2x) is the Catalan function.
a(n) = upper left term in M^n, where M is an infinite square production matrix in which two columns of (1,2,3,...) are prepended to an infinite lower triangular matrix of all 1's and the rest zeros, as follows:
1, 1, 0, 0, 0, 0, ...
2, 2, 1, 0, 0, 0, ...
3, 3, 1, 1, 0, 0, ...
4, 4, 1, 1, 1, 0, ...
5, 5, 1, 1, 1, 1, ...
... (End)
Given g.f. A(x), then 0 = A(x)^2 * (x^3 - 2*x^2 + 5*x - 1) + A(x) *(x^2 - 9*x + 2) + (x^2 + 4*x -1). - Michael Somos, Jan 14 2014
0 = a(n)*(16*a(n+1) +6*a(n+2) -14*a(n+3) +210*a(n+4) -128*a(n+5) +18*a(n+6)) +a(n+1)*(-46*a(n+1) +143*a(n+2) -173*a(n+3) -283*a(n+4) +202*a(n+5) -29*a(n+6)) +a(n+2)*(-63*a(n+2) +386*a(n+3) +765*a(n+4) -529*a(n+5) +75*a(n+6)) +a(n+3)*(-559*a(n+3) +509*a(n+4) -149*a(n+5) +19*a(n+6)) +a(n+4)*(-108*a(n+4) +71*a(n+5) -12*a(n+6)) +a(n+5)*(-4*a(n+5) +a(n+6)). - Michael Somos, Jan 14 2014
G.f.: ( 2 - 9*x + x^2 + (x + x^2) * sqrt(1 - 4*x) ) / (2 - 10*x + 4*x^2 - 2*x^3). - Michael Somos, Apr 15 2012
G.f. = (1 - 3*y + y^2) / (1 - 4*y + 3*y^2 - y^3) = 1 / (1 - y / (1 - y / (1 - 2*y / (1 + y / (2 - y))))) where y = (1 - sqrt(1 - 4*x)) / 2. - Michael Somos, Apr 12 2012
D-finite with recurrence (-n+1)*a(n) +4*(2*n-3)*a(n-1) +(-13*n+19)*a(n-2) +(-13*n+75)*a(n-3) +(5*n-29)*a(n-4) +2*(-2*n+9)*a(n-5)=0. - R. J. Mathar, Jul 27 2013
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EXAMPLE
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G.f. = 1 + x + 3*x^2 + 12*x^3 + 52*x^4 + 232*x^5 + 1049*x^6 + 4777*x^7 + 21845*x^8 + ...
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MAPLE
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C:=(1-sqrt(1-4*x))/2/x: G:=(2-(1+x)*C)/(2-x-(1+x)*C): Gser:=series(G, x=0, 30): seq(coeff(Gser, x, n), n=0..26);
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (2 - 9 x + x^2 + (x + x^2) Sqrt[1 - 4 x]) / (2 (1 - 5 x + 2 x^2 - x^3)), {x, 0, n}]; (* Michael Somos, Jan 14 2014 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( (2 - 9*x + x^2 + (x + x^2) * sqrt(1 - 4*x + x * O(x^n)) ) / (2 * (1 - 5*x + 2*x^2 - x^3)), n))}; /* Michael Somos, Jan 14 2014 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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