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 A227035 Sum(binomial(n,4*k)*binomial(5*k,k)/(4*k+1), k=0..floor(n/4)). 3
 1, 1, 1, 1, 2, 6, 16, 36, 76, 172, 436, 1156, 3006, 7606, 19202, 49466, 130156, 345356, 915196, 2421532, 6427001, 17163581, 46087911, 124133531, 334850208, 904691576, 2449891276, 6651540676, 18100561856, 49344295152, 134719523056, 368350942416, 1008680051756 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Generally, Sum(binomial(n,p*k)*binomial((p+1)*k,k)/(p*k+1), k=0..floor(n/p)) is asymptotic to (p+(p+1)^(1+1/p))^(n+3/2)/(p^(n+1)*(p+1)^(1+3/(2*p))*n^(3/2)*sqrt(2*Pi)) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 FORMULA Recurrence: -2869*(n-7)*(n-6)*(n-5)*(n-4)*a(n-8) + 2*(n-6)*(n-5)*(n-4)*(5226*n-17267)*a(n-7) - (n-5)*(n-4)*(11582*n^2-55156*n+50139)*a(n-6) - 3*(n-4)*(612*n^3 - 18926*n^2 + 102684*n - 155665)*a(n-5) + 5*(n-4)*(2959*n^3 - 26172*n^2 + 77408*n - 76800)*a(n-4) - 1024*(n-2)*(2*n-5)*(7*n^2-35*n+48)*a(n-3) + 1024*(n-2)*(n-1)*(7*n^2-28*n+30)*a(n-2) - 1024*(n-2)*(n-1)*n*(2*n-3)*a(n-1) + 256*(n-2)*(n-1)*n*(n+1)*a(n) = 0. a(n) ~ (4+5^(1+1/4))^(n+3/2)/(4^(n+1)*5^(1+3/8)*n^(3/2)*sqrt(2*Pi)). MATHEMATICA Table[Sum[Binomial[n, 4*k]*Binomial[5*k, k]/(4*k+1), {k, 0, Floor[n/4]}], {n, 0, 20}] PROG (PARI) a(n)=sum(k=0, n\4, binomial(n, 4*k)*binomial(5*k, k)/(4*k+1)) \\ Charles R Greathouse IV, Jun 28 2013 CROSSREFS Cf. A007317 (p=1), A049130 (p=2), A226974 (p=3), A226910 (p=5) Sequence in context: A157136 A178523 A270810 * A257198 A053210 A066641 Adjacent sequences:  A227032 A227033 A227034 * A227036 A227037 A227038 KEYWORD nonn AUTHOR Vaclav Kotesovec, Jun 28 2013 STATUS approved

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)