OFFSET
0,4
COMMENTS
Number of positive terms in expansion of (x_1+x_2+...+x_{n-1}-x_n)^(n+1). - Sergio Falcon, Feb 08 2007
Without the beginning "1" and "-1", we obtain the second diagonal over the principal diagonal of the array notified by B. Cloitre in A026641 and used by R. Choulet in A172025, and from A172061 to A172066. - Richard Choulet, Jan 25 2010
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
From Richard Choulet, Jan 25 2010: (Start)
G.f: f such as: f(z)=(2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^(-2).
a(n) = Sum_{j=0..n+2} (-1)^j*binomial(2*n-j+2, 2+n-j). (End)
Recurrence: 2*n*(3*n-7)*a(n) = (21*n^2 - 61*n + 48)*a(n-1) + 2*(2*n-3)*(3*n-4)*a(n-2). - Vaclav Kotesovec, Apr 19 2014
a(n) ~ 2^(2*n-1)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014
MAPLE
for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+2, 2+n-p)', p=0..n+2): od:seq(a(n), n=0..40):od; taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^(-2), z=0, 42); # Richard Choulet, Jan 25 2010
MATHEMATICA
Table[Sum[Binomial[n+i-2, i]*(-1)^(n-i), {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 19 2014 *)
Table[(-1)^n 2^(1-n)+Binomial[-1+2 n, 1+n] Hypergeometric2F1[1, 2 n, 2+n, -1], {n, 0, 20}] (* Vaclav Kotesovec, Apr 19 2014 *)
With[{k = -2}, CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1 - Sqrt[1-4*x])/(2*x))^k, {x, 0, 30}], x]] (* G. C. Greubel, Feb 18 2019 *)
PROG
(PARI) a(n)=sum(i=0, n, binomial(n+i-2, i)*(-1)^(n-i));
(PARI) k=-2; my(x='x+O('x^30)); Vec((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k) \\ G. C. Greubel, Feb 18 2019
(Magma) k:=-2; m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (2/(3*Sqrt(1-4*x)-1+4*x))*((1-Sqrt(1-4*x))/(2*x))^k )); // G. C. Greubel, Feb 18 2019
(Sage) k=-2; ((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 18 2019
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jan 18 2004
STATUS
approved