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 A118971 a(n) = binomial(5*n+3,n)/(n+1). 23
 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 18974357220, 205263418941, 2240623268512, 24648785802336, 272994644359580, 3041495503591365, 34064252968167180, 383302465665133014 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A quadrisection of A118968. For n >= 1, a(n-1) is the number of lattice paths from (0,0) to (4n,n) using only the steps (1,0) and (0,1) and which stay strictly below the line y = x/4 except at the path's endpoints. - Lucas A. Brown, Aug 21 2020 This is instance k = 4 of the family {c(k, n+1)}_{n>=0} given in a comment in A130564. - Wolfdieter Lang, Feb 04 2024 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..924 Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020. Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019. J. Sawada, J. Sears, A. Trautrim, and A. Williams, Demystifying our Grandparent's De Bruijn Sequences with Concatenation Trees, arXiv:2308.12405 [math.CO], 2023. FORMULA G.f.: If the inverse series of y*(1-y)^4 is G(x) then A(x)=G(x)/x. D-finite with recurrence 8*(4*n+1)*(2*n+1)*(4*n+3)*(n+1)*a(n) -5*(5*n+1)*(5*n+2)*(5*n+3)*(5*n-1)*a(n-1)=0. - R. J. Mathar, Nov 26 2012 a(n) = (4/5)*binomial(5*(n+1),n+1)/(5*(n+1)-1). - Bruno Berselli, Jan 17 2014 E.g.f.: 4F4(4/5,6/5,7/5,8/5; 5/4,3/2,7/4,2; 3125*x/256). - Ilya Gutkovskiy, Jan 23 2018 G.f.: 5F4([4,5,6,7,8]/5, [5,6,7,8]/4; (5^5/4^4)*x) = (4/(5*x))*(1 - 4F3([-1,1,2,3]/5, [1,2,3]/4; (5^5/4^4)*x)). - Wolfdieter Lang, Feb 15 2024 MATHEMATICA Table[4*Binomial[5n+3, n]/(4n+4), {n, 0, 30}] (* Harvey P. Dale, Apr 09 2012 *) CROSSREFS Cf. A000108, A006013, A006632, A130564, A130565, A234466, A234513, A234573, A235340 (members of the same family). Sequence in context: A253255 A141381 A369124 * A369102 A124554 A321586 Adjacent sequences: A118968 A118969 A118970 * A118972 A118973 A118974 KEYWORD easy,nonn AUTHOR Paul Barry, May 07 2006 STATUS approved

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Last modified June 17 11:50 EDT 2024. Contains 373445 sequences. (Running on oeis4.)