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A349290
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G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(x)^4)).
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11
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1, 2, 11, 96, 1001, 11456, 139013, 1756596, 22867421, 304560171, 4130200726, 56836946342, 791689962811, 11140615233281, 158140107648676, 2261708608884896, 32559326010349817, 471428798399646336, 6860801662510005266, 100302910051255600486
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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) = Sum_{k=0..n} binomial(n+3*k,4*k) * binomial(5*k,k) / (4*k+1).
a(n) = F([1/5, 2/5, 3/5, 4/5, (1+n)/3, (2+n)/3, (3+n)/3, -n], [1/4, 1/2, 1/2, 3/4, 3/4, 1, 5/4], -3^3*5^5/2^16), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 13 2021
a(n) ~ sqrt(1 + 3*r) / (2 * 5^(3/4) * sqrt(2*Pi*(1-r)) * n^(3/2) * r^(n + 1/4)), where r = 0.0631152861998150860738633360987635931... is the root of the equation 5^5 * r = 4^4 * (1-r)^4. - Vaclav Kotesovec, Nov 14 2021
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MATHEMATICA
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nmax = 19; A[_] = 0; Do[A[x_] = 1/((1 - x) (1 - x A[x]^4)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 3 k, 4 k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 19}]
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(n+3*k, 4*k) * binomial(5*k, k) / (4*k+1)); \\ Michel Marcus, Nov 14 2021
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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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