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A182447
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a(n) = Sum_{k = 0..n} C(n,k)^10.
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12
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1, 2, 1026, 118100, 62563330, 20019531252, 11393421713604, 5550455033938152, 3431955863873102850, 2052124795850957537060, 1367610300690018553312276, 916694195766256069610158152, 649630217578404016288230718276, 467800319852823195772146025385000
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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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Asymptotic (p = 10): a(n) ~ 2^(p*n)/sqrt(p)*(2/(Pi*n))^((p - 1)/2)*( 1 - (p - 1)^2/(4*p*n) + O(1/n^2) ).
For r a nonnegative integer, Sum_{k = r..n} C(k,r)^10*C(n,k)^10 = C(n,r)^10*a(n-r), where we take a(n) = 0 for n < 0. - Peter Bala, Jul 27 2016
Sum_{n>=0} a(n) * x^n / (n!)^10 = (Sum_{n>=0} x^n / (n!)^10)^2. - Ilya Gutkovskiy, Jul 17 2020
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MAPLE
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a := n -> hypergeom([seq(-n, i=1..10)], [seq(1, i=1..9)], 1):
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MATHEMATICA
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Table[Sum[Binomial[n, k]^10, {k, 0, n}], {n, 0, 25}]
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(n, k)^10); \\ Michel Marcus, Jul 17 2020
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CROSSREFS
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Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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