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 A337370 Expansion of sqrt(2 / ( (1-12*x+4*x^2) * (1-2*x+sqrt(1-12*x+4*x^2)) )). 4
 1, 8, 74, 736, 7606, 80464, 864772, 9400192, 103061158, 1137528688, 12623082284, 140697113792, 1574005263676, 17663830073504, 198760191043784, 2241743315230208, 25335473017856774, 286850379192127664, 3252960763923781276, 36942512756224955456, 420084161646913792724 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Robert Israel, Table of n, a(n) for n = 0..938 FORMULA a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k). 8*(2*n - 3)*(n - 2)*a(n - 3) - 4*(10*n^2 - 35*n + 27)*a(n - 2) - 2*(10*n^2 + 5*n - 3)*a(n - 1) + (2*n + 1)*n*a(n) = 0. - Robert Israel, Aug 27 2020 a(0) = 1, a(1) = 8 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * (24*n^2-12*n-4) * a(n-1) - 4 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020 a(n) ~ 2^(n - 5/4) * (1 + sqrt(2))^(2*n + 3/2) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 31 2020 MAPLE Rec:= 8*(2*n - 3)*(n - 2)*a(n - 3) - 4*(10*n^2 - 35*n + 27)*a(n - 2) - 2*(10*n^2 + 5*n - 3)*a(n - 1) + (2*n + 1)*n*a(n) = 0: f:= gfun:-rectoproc({Rec, a(0)=1, a(1)=8, a(2)=74}, a(n), remember): map(f, [\$0..30]); # Robert Israel, Aug 27 2020 MATHEMATICA a[n_] := Sum[2^(n - k) * Binomial[2*k, k] * Binomial[2*n + 1, 2*k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Aug 25 2020 *) PROG (PARI) N=40; x='x+O('x^N); Vec(sqrt(2/((1-12*x+4*x^2)*(1-2*x+sqrt(1-12*x+4*x^2))))) (PARI) {a(n) = sum(k=0, n, 2^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))} CROSSREFS Column k=2 of A337369. Cf. A337390. Sequence in context: A190953 A220458 A144850 * A357208 A163970 A116251 Adjacent sequences: A337367 A337368 A337369 * A337371 A337372 A337373 KEYWORD nonn AUTHOR Seiichi Manyama, Aug 25 2020 STATUS approved

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Last modified August 14 12:42 EDT 2024. Contains 375164 sequences. (Running on oeis4.)