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A051714
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Numerators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)*(a(n,k) - a(n,k+1)), n >= 0, k >= 0.
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22
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1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 3, 1, -1, 1, 1, 2, 1, -1, 0, 1, 1, 5, 2, -3, -1, 1, 1, 1, 3, 5, -1, -1, 1, 0, 1, 1, 7, 5, 0, -4, 1, 1, -1, 1, 1, 4, 7, 1, -1, -1, 1, -1, 0, 1, 1, 9, 28, 49, -29, -5, 8, 1, -5, 5, 1, 1, 5, 3, 8, -7, -9, 5, 7, -5, 5, 0, 1, 1, 11, 15, 27, -28, -343, 295, 200, -44, -1017, 691, -691
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OFFSET
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0,13
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COMMENTS
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LINKS
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FORMULA
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a(n,k) = numerator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)/(j+k+1)).
E.g.f.: A(x,t) = (x+log(1-t))/(1-t-exp(-x)) = (1+(1/2)*x+(1/6)*x^2/2!-(1/30)*x^4/4!+...)*1 + (1/2+(1/3)*x+(1/6)*x^2/2!+...)*t + (1/3+(1/4)*x+(3/20)*x^2/2!+...)*t^2 + .... (End)
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EXAMPLE
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Table begins:
1 1/2 1/3 1/4 1/5 1/6 1/7 ...
1/2 1/3 1/4 1/5 1/6 1/7 ...
1/6 1/6 3/20 2/15 5/42 ...
0 1/30 1/20 2/35 5/84 ...
-1/30 -1/30 -3/140 -1/105 ...
Antidiagonals of numerator(a(n,k)):
1;
1, 1;
1, 1, 1;
1, 1, 1, 0;
1, 1, 3, 1, -1;
1, 1, 2, 1, -1, 0;
1, 1, 5, 2, -3, -1, 1;
1, 1, 3, 5, -1, -1, 1, 0;
1, 1, 7, 5, 0, -4, 1, 1, -1;
1, 1, 4, 7, 1, -1, -1, 1, -1, 0;
1, 1, 9, 28, 49, -29, -5, 8, 1, -5, 5;
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MAPLE
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a:= proc(n, k) option remember;
`if`(n=0, 1/(k+1), (k+1)*(a(n-1, k)-a(n-1, k+1)))
end:
seq(seq(numer(a(n, d-n)), n=0..d), d=0..12); # Alois P. Heinz, Apr 17 2013
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MATHEMATICA
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nmax = 12; a[0, k_]:= 1/(k+1); a[n_, k_]:= a[n, k]= (k+1)(a[n-1, k]-a[n-1, k+1]); Numerator[Flatten[Table[a[n-k, k], {n, 0, nmax}, {k, n, 0, -1}]]] (* Jean-François Alcover, Nov 28 2011 *)
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PROG
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(Magma)
function a(n, k)
if n eq 0 then return 1/(k+1);
else return (k+1)*(a(n-1, k) - a(n-1, k+1));
end if;
end function;
A051714:= func< n, k | Numerator(a(n, k)) >;
(SageMath)
def a(n, k):
if (n==0): return 1/(k+1)
else: return (k+1)*(a(n-1, k) - a(n-1, k+1))
def A051714(n, k): return numerator(a(n, k))
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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