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A078995
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a(n) = Sum_{k=0..n} C(4*k,k)*C(4*(n-k),n-k).
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8
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1, 8, 72, 664, 6184, 57888, 543544, 5113872, 48180456, 454396000, 4288773152, 40503496536, 382701222296, 3617396099936, 34203591636048, 323492394385824, 3060238763412072, 28955508198895584, 274018698082833760, 2593539713410178528, 24550565251665845664
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OFFSET
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0,2
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LINKS
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D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Y_n for s=4).
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FORMULA
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a(n) = 2/3*(256/27)^n*(1+c/sqrt(n)+o(n^-1/2)) where c = 2/3*sqrt(2/(3*Pi)) = 0.307105910641187... More generally, a(n, m)=sum(k=0, n, binomial(m*k, k)*binomial(m*(n-k), n-k)) is asymptotic to 1/2*m/(m-1)*(m^m/(m-1)^(m-1))^n. See A000302, A006256 for cases m=2 and 3. - Benoit Cloitre, Jan 26 2003, corrected and extended by Vaclav Kotesovec, Nov 06 2012
243*n*(8*n - 17)*(3*n - 1)*(3*n - 4)*(3*n - 2)*(3*n - 5)*a(n) = 72*(3*n - 5)*(3*n - 4)*(6912*n^4 - 33120*n^3 + 58256*n^2 - 47798*n + 15309)*a(n - 1) - 3072*(2*n - 3)*(6912*n^5 - 55008*n^4 + 175696*n^3 - 282180*n^2 + 227825*n - 73710)*a(n - 2) + 262144*(n - 2)*(4*n - 7)*(2*n - 3)*(2*n - 5)*(4*n - 9)*(8*n - 9)*a(n - 3). - Vladeta Jovovic, Jul 16 2004
Shorter recurrence: 81*n*(3*n-2)*(3*n-1)*(8*n-11)*a(n) = 24*(4608*n^4-14400*n^3+15776*n^2-7346*n+1215)*a(n-1) - 2048*(2*n-3)*(4*n-5)*(4*n-3)*(8*n-3)*a(n-2). - Vaclav Kotesovec, Nov 06 2012
a(n) = sum(k=0,n,binomial(4*k+l,k)*binomial(4*(n-k)-l,n-k)) for every real number l. - Rui Duarte and António Guedes de Oliveira, Feb 16 2013
a(n) = sum(k=0,n,3^(n-k)*binomial(4n+1,k)). - Rui Duarte and António Guedes de Oliveira, Feb 17 2013
a(n) =sum(k=0,n,4^(n-k)*binomial(3n+k,k)). - Rui Duarte and António Guedes de Oliveira, Feb 17 2013
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MAPLE
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series(eval(g/(3*g-4), g=RootOf(g = 1+x*g^4, g))^2, x=0, 30); # Mark van Hoeij, May 06 2013
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MATHEMATICA
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Table[Sum[Binomial[4*k, k]*Binomial[4*(n - k), n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 06 2012 *)
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(4*k, k)*binomial(4*(n-k), n-k)); \\ Michel Marcus, May 09 2020
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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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