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A300480
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Rectangular array read by antidiagonals: a(m,n) = 2 * Integral_{t>=0} T_n((t+m)/2)*exp(-t)*dt, m>=0, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.
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7
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2, 2, 1, 2, 2, 0, 2, 3, 3, 3, 2, 4, 8, 10, 18, 2, 5, 15, 29, 47, 95, 2, 6, 24, 66, 130, 256, 592, 2, 7, 35, 127, 327, 697, 1610, 4277, 2, 8, 48, 218, 722, 1838, 4376, 11628, 35010, 2, 9, 63, 345, 1423, 4459, 11770, 31607, 95167, 320589, 2, 10, 80, 514, 2562, 9820, 30248, 85634, 258690
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OFFSET
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0,1
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COMMENTS
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a(m,n) is a polynomial in m of degree n.
For any integers m>=0, n>=0, 2 * Integral_{t=-m..m} T_n(t/2)*exp(-t)*dt = 4 * Integral_{z=-m/2..m/2} T_n(z)*exp(-2*z)*dz = A300481(m,n)*exp(m) - a(m,n)*exp(-m).
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LINKS
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FORMULA
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a(m,n) = Sum_{i=0..n} A127672(n,i) * i! * Sum_{j=0..i} m^j/j!.
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EXAMPLE
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Array starts with:
m=0: 2, 1, 0, 3, 18, 95, 592, ...
m=1: 2, 2, 3, 10, 47, 256, 1610, ...
m=2: 2, 3, 8, 29, 130, 697, 4376, ...
m=3: 2, 4, 15, 66, 327, 1838, 11770, ...
m=4: 2, 5, 24, 127, 722, 4459, 30248, ...
...
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PROG
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(PARI) { A300480(m, n) = if(n==0, return(2)); subst( serlaplace( 2*polchebyshev(n, 1, (x+m)/2)), x, 1); }
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CROSSREFS
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Values for m<=0 are given in A300481.
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KEYWORD
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AUTHOR
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STATUS
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approved
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