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A349293 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(x)^7)). 11
1, 2, 17, 249, 4345, 83285, 1694273, 35915349, 784691569, 17545398747, 399545961817, 9234298584921, 216053290499201, 5107287712887563, 121795876378121121, 2926604574330886897, 70788399943851406825, 1722188546498276868124, 42114624858397590035177 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
In general, for k>=1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial((k+1)*j,j) / (k*j+1) ~ sqrt(1 + (k-1)*r) / ((k+1)^(1/k) * sqrt(2*k*(k+1)*Pi*(1-r)) * n^(3/2) * r^(n + 1/k)), where r is the smallest real root of the equation (k+1)^(k+1) * r = k^k * (1-r)^k. - Vaclav Kotesovec, Nov 14 2021
LINKS
FORMULA
a(n) = Sum_{k=0..n} binomial(n+6*k,7*k) * binomial(8*k,k) / (7*k+1).
a(n) ~ sqrt(1 + 6*r) / (2^(17/7) * sqrt(7*Pi*(1-r)) * n^(3/2) * r^(n + 1/7)), where r = 0.0375502499742240443056934699070050852345109331376051496159609551... is the real root of the equation 8^8 * r = 7^7 * (1-r)^7. - Vaclav Kotesovec, Nov 14 2021
MATHEMATICA
nmax = 18; A[_] = 0; Do[A[x_] = 1/((1 - x) (1 - x A[x]^7)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 6 k, 7 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]
PROG
(PARI) a(n) = sum(k=0, n, binomial(n+6*k, 7*k) * binomial(8*k, k) / (7*k+1)); \\ Michel Marcus, Nov 14 2021
CROSSREFS
Sequence in context: A099694 A099698 A218681 * A098622 A020561 A099702
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 13 2021
STATUS
approved

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Last modified April 21 10:39 EDT 2024. Contains 371852 sequences. (Running on oeis4.)