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A082395
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Number of shifted Young tableaux with height <= 3.
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7
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1, 1, 2, 3, 6, 12, 27, 63, 154, 386, 989, 2574, 6787, 18085, 48622, 131719, 359194, 985186, 2715973, 7521568, 20915257, 58373587, 163462816, 459136810, 1293223231, 3651864607, 10336625732, 29321683083, 83344398534, 237344961292
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n}(-1)^(k+1)*binomial(n, k)*binomial(k-1, floor(k/2)). - Vladeta Jovovic, Sep 18 2003
Recurrence: 3*(n-2)*a(n-3)+(2-n)*a(n-2)+(4-3*n)*a(n-1)+n*a(n)=0. - Vaclav Kotesovec, Oct 02 2012
Asymptotic: a(n) ~ 3^(n+3/2)/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 02 2012
G.f. A(x) = (1 + x - sqrt(1 - 2*x - 3*x^2))/(2*(1-x^2)).
G.f. A(x) = A(x)^2 + (1 - A(x)^2)*x + (A(x) - A(x)^2)*x^2 + A(x)^2*x^3.
G.f. A(x) satisfies 0 = Sum_{n>=0} (-1)^n * x^(n*(n-3)/2) * A(x)^n / Product_{k=0..n+1} (1 - x^k*A(x)). (End)
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EXAMPLE
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G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 27*x^7 + 63*x^8 + 154*x^9 + 386*x^10 + 989*x^11 + 2574*x^12 + ...
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MATHEMATICA
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Table[Sum[(-1)^(k+1)*Binomial[n, k]*Binomial[k-1, Floor[k/2]], {k, 1, n}], {n, 1, 20}]
RecurrenceTable[{3*(n-2)*a[n-3]+(2-n)*a[n-2]+(4-3n)*a[n-1]+n*a[n]==0, a[1]==1, a[2]==1, a[3]==2}, a, {n, 20}] (* Vaclav Kotesovec, Oct 02 2012 *)
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PROG
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(Sage)
a, b, s, n = 1, 0, 1, 1
yield a
while True:
s += b
yield s
n += 1
a, b = b, (2*b+3*a)*(n-1)/(n+1)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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