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 A172062 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=5. 7
 1, 6, 29, 128, 541, 2232, 9076, 36568, 146446, 584082, 2322967, 9220544, 36548573, 144732176, 572756312, 2265577184, 8959034798, 35421613196, 140035644602, 553606049024, 2188652065586, 8653317051056, 34216118389384 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This sequence is the 5th 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+5 that are 3 steps higher than the preceding peak. - David Scambler, Apr 22 2013 Apparently half the sum of all height differences between adjacent peaks in all Dyck paths of semilength n+3. - David Scambler, Apr 22 2013 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA a(n)=sum('(-1)^(p)*binomial(2*n+k-p,n-p)',p=0..n) (with k=5). a(n) ~ 2^(2*n+6)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014 Conjecture: 2*n*(n+5)*(3*n+7)*a(n) -(n+3)*(21*n^2+79*n+80)*a(n-1) -2*(3*n+10)*(2*n+3)*(n+2)*a(n-2)=0. - R. J. Mathar, Feb 19 2016 EXAMPLE a(4) = C(13,4) - C(12,3) + C(11,2) - C(10,1) + C(9,0) = 13*11*5 - 20*11 + 55 - 10 + 1 = 541. 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; 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))^5, {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2014 *) CROSSREFS Sequence in context: A225618 A081278 A054146 * A081674 A173413 A008549 Adjacent sequences:  A172059 A172060 A172061 * A172063 A172064 A172065 KEYWORD easy,nonn AUTHOR Richard Choulet, Jan 24 2010 STATUS approved

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Last modified October 16 02:45 EDT 2018. Contains 316252 sequences. (Running on oeis4.)