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A080956
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a(n) = (n+1)*(2-n)/2.
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34
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1, 1, 0, -2, -5, -9, -14, -20, -27, -35, -44, -54, -65, -77, -90, -104, -119, -135, -152, -170, -189, -209, -230, -252, -275, -299, -324, -350, -377, -405, -434, -464, -495, -527, -560, -594, -629, -665, -702, -740, -779, -819, -860, -902, -945, -989, -1034, -1080, -1127, -1175, -1224, -1274, -1325, -1377
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OFFSET
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0,4
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COMMENTS
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Coefficient of x in the polynomial C(n,0)+C(n+1,1)x+C(n+2,2)x(x-1)/2.
a(n) is essentially the case 1 of the polygonal numbers. The polygonal numbers are defined as P_k(n) = Sum_{i=1..n} ((k-2)*i-(k-3)). Thus P_1(n) = n*(3-n)/2 and a(n) = P_1(n+1). See A005563 for the case k=0. - Peter Luschny, Jul 08 2011
This is the case k=-1 of the formula (k*m*(m+1)-(-1)^k+1)/2. See similar sequences listed in A262221. - Bruno Berselli, Sep 17 2015
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LINKS
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FORMULA
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a(n) = 2*(C(n+1, 1)-C(n+2, 2)) = (n+1)*(2-n)/2.
If we define f(n,i,a) = Sum_{k=0..n-i} (binomial(n,k)*stirling1(n-k,i)*Product_{j=0..k-1} (-a-j)), then a(n) = f(n,n-1,2), for n>=3. - Milan Janjic, Dec 20 2008
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EXAMPLE
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MAPLE
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G(x):=exp(x)*(x-x^2/2): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=1..54 ); # Zerinvary Lajos, Apr 05 2009
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {1, 1, 0}, 60] (* Harvey P. Dale, Nov 29 2019 *)
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PROG
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(PARI) a(n)=(n+1)*(2-n)/2;
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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Lajos e.g.f. adapted to offset zero by R. J. Mathar, Jun 11 2009
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STATUS
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approved
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