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 A172064 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=7. 11
 1, 8, 46, 230, 1068, 4744, 20476, 86662, 361711, 1494384, 6126818, 24972326, 101320712, 409609664, 1651162688, 6640469816, 26655382802, 106830738224, 427612715516, 1709790470780, 6830461107736, 27266848437608 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This sequence is the 7th 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. Apparently the number of peaks in all Dyck paths of semilength n+7 that are 5 steps higher than the preceding peak. - David Scambler, Apr 22 2013 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=7. a(n) ~ 2^(2*n+8)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014 Conjecture: 2*n*(n+7)*(3*n+11)*a(n) -(21*n^3+212*n^2+719*n+840)*a(n-1) -2*(2*n+5)*(n+3)*(3*n+14)*a(n-2)=0. - R. J. Mathar, Feb 19 2016 EXAMPLE a(4) = C(15,4) - C(14,3) + C(13,2) - C(12,1) + C(11,0) = 7*13*15 - 14*13*2 + 78 - 12 + 1 = 1068. MAPLE for k from 0 to 20 do for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+k, n-p)', p=0..n): od:seq(a(n), n=0..40):od; # 2nd program for k from 0 to 40 do taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k, z=0, 40+k):od; MATHEMATICA CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1-Sqrt[1-4*x])/(2*x))^7, {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2014 *) PROG (PARI) k=7; 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:=7; 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=7; ((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 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), A172065 (k=8), A172066 (k=9), A172067 (k=10). Sequence in context: A134114 A071586 A027650 * A197238 A182542 A026843 Adjacent sequences:  A172061 A172062 A172063 * A172065 A172066 A172067 KEYWORD easy,nonn AUTHOR Richard Choulet, Jan 24 2010 STATUS approved

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Last modified August 3 13:59 EDT 2021. Contains 346438 sequences. (Running on oeis4.)