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 A172065 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=8. 11
 1, 9, 56, 297, 1444, 6656, 29618, 128603, 548591, 2309467, 9624964, 39799813, 163556776, 668796712, 2723729944, 11055878188, 44753742226, 180746332690, 728571706240, 2932018571370, 11783070278816, 47297147250204 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This sequence is the 8th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The Maple programs give the first diagonals of this array. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA a(n) = Sum_{j=0..n} (-1)^j *binomial(2*n+k-j, n-j), with k=8. a(n) ~ 2^(2*n+9)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014 Conjecture: 2*n*(n+8)*(3*n+13)*a(n) -(21*n^3 + 247*n^2 + 980*n + 1344)*a(n-1) - 2*(n+3)*(3*n+16)*(2*n+7)*a(n-2) = 0. - R. J. Mathar, Feb 29 2016 EXAMPLE a(4) = C(16,4) - C(15,3) + C(14,2) - C(13,1) + C(12,0) = 20*91 - 35*13 + 91 - 13 + 1 = 1820 - 455 + 79 = 1444. MAPLE a:= n-> add((-1)^(p)*binomial(2*n-p+8, n-p), p=0..n): seq(a(n), n=0..40); # 2nd program a:= n-> coeff(series((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))         /(2*z))^8, z, n+20), z, n): seq(a(n), n=0..40); MATHEMATICA CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1-Sqrt[1-4*x])/(2*x))^8, {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2014 *) PROG (PARI) k=8; 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 17 2019 (MAGMA) k:=8 m:=30; R:=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 17 2019 (Sage) k=8; m=30; a=((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k ).series(x, m+2).coefficients(x, sparse=False); a[0:m] # G. C. Greubel, Feb 17 2019 CROSSREFS Cf. A091526 (k=-2), A072547 (k=-1), A026641 (k=0), A014300 (k=1), A014301 (k=2), A172025 (k=3), A172061 (k=4), A172062 (k=5), A172063 (k=6), A172064 (k=7), A172066 (k=9), A172067 (k=10) Sequence in context: A097556 A196861 A211844 * A002055 A026842 A026846 Adjacent sequences:  A172062 A172063 A172064 * A172066 A172067 A172068 KEYWORD easy,nonn AUTHOR Richard Choulet, Jan 24 2010 STATUS approved

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Last modified December 9 00:32 EST 2019. Contains 329871 sequences. (Running on oeis4.)