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 A349361 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^5 / (1 + x). 11
 1, 1, 4, 26, 194, 1581, 13625, 122120, 1126780, 10631460, 102104845, 994855179, 9809872626, 97710157154, 981636609906, 9935473707279, 101214412755647, 1036991125300748, 10678412226507032, 110459290208905008, 1147261657267290037 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..500 FORMULA a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(5*k,k) / (4*k+1). a(n) = (-1)^(n+1)*F([6/5, 7/5, 8/5, 9/5, 1-n], [3/2, 7/4, 2, 9/4], 5^5/2^8), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 15 2021 From Vaclav Kotesovec, Nov 17 2021: (Start) a(n) ~ 2869^(n + 1/2) / (25 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). Recurrence: 8*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*a(n) = 3*(615*n^4 - 718*n^3 - 275*n^2 + 618*n - 200)*a(n-1) + 4*(n-2)*(2485*n^3 - 6879*n^2 + 6524*n - 2040)*a(n-2) + 2*(n-3)*(n-2)*(8095*n^2 - 23517*n + 18092)*a(n-3) + 12*(n-4)*(n-3)*(n-2)*(935*n - 1838)*a(n-4) + 2869*(n-5)*(n-4)*(n-3)*(n-2)*a(n-5). (End) MAPLE a:= n-> coeff(series(RootOf(1+x*A^5/(1+x)-A, A), x, n+1), x, n): seq(a(n), n=0..20); # Alois P. Heinz, Nov 15 2021 MATHEMATICA nmax = 20; A[_] = 0; Do[A[x_] = 1 + x A[x]^5/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}] CROSSREFS Cf. A001006, A002294, A127897, A317133, A345368 (binomial transform), A346665, A349332, A349362, A349363, A349364. Sequence in context: A127086 A198024 A278393 * A192499 A271935 A246509 Adjacent sequences: A349358 A349359 A349360 * A349362 A349363 A349364 KEYWORD nonn AUTHOR Ilya Gutkovskiy, Nov 15 2021 STATUS approved

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Last modified June 23 22:09 EDT 2024. Contains 373661 sequences. (Running on oeis4.)