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A277188
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The binomial sum a(n) = Sum_{k=0..n}(binomial(n,k)*binomial(n+1,k+1)*binomial(n+2,k+2)).
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1
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1, 7, 43, 281, 1896, 13112, 92359, 659941, 4769758, 34797170, 255838760, 1893389720, 14091400480, 105385445856, 791504226943, 5966958725021, 45133376297922, 342400478465678, 2604549070175770, 19860078537996958, 151769147958738016
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = (1/2)*(n+1)^2*(n+2)*hypergeometric({-n,-n,-n},{2,3},-1).
Recurrence: (n+2)^2*(6*n^3 + 3*n^2 - 3*n - 2)*a(n) = (42*n^5 + 147*n^4 + 147*n^3 - 8*n^2 - 60*n - 16)*a(n-1) + 8*(n-1)*n*(6*n^3 + 21*n^2 + 21*n + 4)*a(n-2).
a(n) ~ 2^(3*n+4)/(sqrt(3)*Pi*n).
(End)
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MATHEMATICA
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Table[Sum[Binomial[n, k]Binomial[n+1, k+1]Binomial[n+2, k+2], {k, 0, n}], {n, 0, 100}]
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PROG
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(Maxima) makelist((n+1)^2*(n+2)/2*hypergeometric([-n, -n, -n], [2, 3], -1), n, 0, 12);
(PARI) a(n) = sum(k=0, n, (binomial(n, k)*binomial(n+1, k+1)*binomial(n+2, k+2))); \\ Michel Marcus, Oct 04 2016
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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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