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A216947
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Number of 2-colored non-crossing set partitions of [n].
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1
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1, 1, 3, 11, 47, 225, 1173, 6529, 38265, 233795, 1478265, 9619065, 64135965, 436693431, 3028075677, 21335364345, 152466207953, 1103356966359, 8075335006933, 59706844371261, 445545262970505, 3352776605782479, 25424338834654587
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OFFSET
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0,3
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LINKS
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FORMULA
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Marberg gives a recurrence and g.f.
a(n) = (2*(5*n^2 + 6*n - 2)*a(n-1) - 9*(n - 2)*(n + 1)*a(n-2))/((n + 2)*(n + 3)) for n >= 2. - Andrew Howroyd, Dec 26 2019, after Marberg theorem 1.7.
a(n) = Sum_{j=0..n} Sum_{k=0..j} C(n-1,j)*C(j+2,k) * C(j+2,k+1) * C(j+2,k+2) ) / (C(j+2,1) * C(j+2,2)). - Michael D. Weiner, Jan 21 2020
G.f.: hypergeom([1/4, 5/4],[2],-64*x/((9*x-1)^3*(x-1)))/(3*(1-9*x)^(3/4)*x^2*(1-x)^(1/4))-(2*x-1)*(x-1)/(3*x^2). - Mark van Hoeij, Jul 27 2021
a(n) = ((n+3)*hypergeom([1/2, -1-n, -2-n],[2, 2],4)-3*(n+2)*hypergeom([1/2, -1-n, -n],[2, 2],4))/(2*n*(n+2)) for n > 0. - Mark van Hoeij, Nov 06 2023
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MAPLE
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option remember ;
local npr ;
if n <=1 then
1 ;
else
npr := n-2 ;
9*npr*(npr+3)*procname(n-2)-2*(5*npr^2+26*npr+30)*procname(n-1) ;
-%/(npr+4)/(npr+5) ;
end if;
end proc:
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MATHEMATICA
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a[n_] := a[n] = Module[{npr, s}, If[n <= 1, 1, npr = n-2; s = 9*npr*(npr + 3)*a[n-2] - 2*(5*npr^2 + 26*npr + 30)*a[n-1]; -s/(npr + 4)/(npr + 5)]];
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PROG
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(PARI) seq(n)={my(a=vector(n+1)); a[1]=1; a[2]=1; for(n=2, n, a[n+1] = (2*(5*n^2 + 6*n - 2)*a[n] - 9*(n - 2)*(n + 1)*a[n-1])/((n + 2)*(n + 3))); a} \\ Andrew Howroyd, Dec 26 2019
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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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