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A359643
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a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k).
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2
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1, 5, 37, 317, 2885, 27105, 259765, 2523813, 24768069, 244941833, 2437083697, 24367722725, 244639635749, 2464477467769, 24899468129405, 252202062544617, 2560119328830725, 26038134699958233, 265278657849511561, 2706809063101138409, 27657194997231516145, 282941098708193905485
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OFFSET
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0,2
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COMMENTS
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In general, for m>1, Sum_{k=0..n} binomial(n,k) * binomial(m*k,k) ~ sqrt((m + (1 - 1/m)^(m-1))/(m-1)) * (1 + m^m/(m-1)^(m-1))^n / sqrt(2*Pi*n).
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LINKS
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FORMULA
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a(n) ~ 283^(n + 1/2) / (2^(7/2) * sqrt(Pi*n) * 3^(3*n + 1/2)).
Conjecture D-finite with recurrence +81*n*(3*n-1)*(3*n-2)*a(n) +3*(243*n^3-8433*n^2+14984*n-7064)*a(n-1) +2*(-58607*n^3+297306*n^2-491401*n+269124)*a(n-2) +6*(n-2)*(56663*n^2-237722*n+252221)*a(n-3) -3*(n-2)*(n-3)*(111625*n-286402)*a(n-4) +110653*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 09 2023
a(n) = 4F3( -n,1/4,1/2,3/4 ; 1/3, 2/3,1 ; -256/27). - R. J. Mathar, Jan 10 2023
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MAPLE
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hypergeom([-n, 1/4, 1/2, 3/4], [1/3, 2/3, 1], -256/27) ;
simplify(%) ;
end proc:
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MATHEMATICA
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Table[Sum[Binomial[n, k]*Binomial[4*k, k], {k, 0, n}], {n, 0, 20}]
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(n, k) * binomial(4*k, k)); \\ Michel Marcus, Jan 09 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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