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A346684
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a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(8*k,k) / (7*k + 1).
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5
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1, 0, 8, 84, 1156, 17122, 268262, 4370086, 73281938, 1256608767, 21933420953, 388400019583, 6960642974905, 126008367913375, 2300862338502425, 42326714610861679, 783717720798538121, 14594469249932149279, 273161824453612674593, 5135931850101477641707
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OFFSET
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0,3
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COMMENTS
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In general, for m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(m*k,k) / ((m-1)*k + 1) ~ m^(m*(n+1) + 1/2) / (sqrt(2*Pi) * (m^m + (m-1)^(m-1)) * n^(3/2) * (m-1)^((m-1)*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 0..768
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FORMULA
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G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^7 * A(x)^8.
a(n) ~ 2^(24*n + 25) / (17600759 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021
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MATHEMATICA
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Table[Sum[(-1)^(n - k) Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; A[_] = 0; Do[A[x_] = 1/(1 + x) + x (1 + x)^7 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
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PROG
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(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(8*k, k)/(7*k + 1)); \\ Michel Marcus, Jul 29 2021
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CROSSREFS
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Cf. A007556, A032357, A188678, A346668, A346672.
Cf. A346680, A346681, A346682, A346683.
Sequence in context: A233835 A300993 A052659 * A350264 A113376 A205311
Adjacent sequences: A346681 A346682 A346683 * A346685 A346686 A346687
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KEYWORD
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nonn
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AUTHOR
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Ilya Gutkovskiy, Jul 29 2021
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STATUS
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approved
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