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 A304941 Expansion of ((1 + 4*x)/(1 - 4*x))^(3/4). 4
 1, 6, 18, 68, 246, 948, 3572, 13896, 53286, 208452, 807132, 3169080, 12346300, 48602760, 190150440, 750018448, 2943363078, 11627329764, 45736940364, 180897649368, 712881236052, 2822389182104, 11138924119512, 44137230865392, 174405194802524, 691557285091176 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Let ((1 + k*x)/(1 - k*x))^(m/k) = a(0) + a(1)*x + a(2)*x^2 + ... then n*a(n) = 2*m*a(n-1) + k^2*(n-2)*a(n-2) for n > 1. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 FORMULA n*a(n) = 6*a(n-1) + 4^2*(n-2)*a(n-2) for n > 1. a(n) ~ 2^(2*n + 3/4) / (Gamma(3/4) * n^(1/4)). - Vaclav Kotesovec, May 28 2018 MATHEMATICA CoefficientList[Series[((1+4x)/(1-4x))^(3/4), {x, 0, 30}], x] (* Harvey P. Dale, Oct 24 2020 *) PROG (PARI) N=66; x='x+O('x^N); Vec(((1+4*x)/(1-4*x))^(3/4)) (Magma) [n le 2 select 6^(n-1) else 2*(3*Self(n-1) + 8*(n-3)*Self(n-2))/(n-1): n in [1..40]]; // G. C. Greubel, Jun 07 2023 (SageMath) @CachedFunction def a(n): # a = A304941 if n<2: return 6^n else: return 2*(3*a(n-1) + 8*(n-2)*a(n-2))//n [a(n) for n in range(41)] # G. C. Greubel, Jun 07 2023 CROSSREFS ((1 + 4*x)/(1 - 4*x))^(m/4): A303537 (m=1), A304940 (m=2), this sequence (m=3), A081654 (m=4). Sequence in context: A034751 A200151 A000623 * A129369 A095853 A027266 Adjacent sequences: A304938 A304939 A304940 * A304942 A304943 A304944 KEYWORD nonn AUTHOR Seiichi Manyama, May 22 2018 STATUS approved

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Last modified September 30 09:43 EDT 2023. Contains 365784 sequences. (Running on oeis4.)