|
| |
|
|
A080956
|
|
(n+1)*(2-n)/2.
|
|
17
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,4
|
|
|
COMMENTS
| Coefficient of x in the polynomial C(n,0)+C(n+1,1)x+C(n+2,2)x(x-1)/2.
Equals A154990 * [1,2,3,...]. [From Gary W. Adamson & Mats Granvik, Jan 19 2009]
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]
|
|
|
LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
|
|
|
FORMULA
| a(n) = 2*(C(n+1, 1)-C(n+2, 2)) = (n+1)*(2-n)/2
G.f.: (1-2*x)/(1-x)^3. [R. J. Mathar, Jun 11 2009]
If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = f(n,n-1,2), for n>=3. [From Milan R. Janjic (agnus(AT)blic.net), Dec 20 2008]
E.g.f.: exp(x)*(1-x^2/2) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009, R. J. Mathar, Jun 11 2009]
|
|
|
MAPLE
| restart: 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 ); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009]
|
|
|
MATHEMATICA
| f[n_]:=n; lst={}; Do[a=f[n]; Do[a-=f[m], {m, n-1, 1, -1}]; AppendTo[lst, a], {n, 30}]; lst [From Vladimir Orlovsky, Feb 10 2010]
|
|
|
PROG
| (MAGMA) [(n+1)*(2-n)/2: n in [0..80]]; // Vincenzo Librandi, Jul 08 2011
(PARI) a(n)=(n+1)*(2-n)/2;
|
|
|
CROSSREFS
| Cf. A000096.
A154990 [From Gary W. Adamson & Mats Granvik, Jan 19 2009]
Sequence in context: A075543 A132315 A132336 * A000096 A132337 A134189
Adjacent sequences: A080953 A080954 A080955 * A080957 A080958 A080959
|
|
|
KEYWORD
| easy,sign
|
|
|
AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Mar 01 2003
|
|
|
EXTENSIONS
| Adapted Lajos e.g.f. to offset zero R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 11 2009
|
| |
|
|
