A349314 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^8) / (1 - x).

1, 2, 18, 274, 4930, 97346, 2039570, 44524818, 1001773058, 23065953794, 540886665618, 12872727013522, 310135678438978, 7549240857128258, 185381380643501970, 4586875745951650706, 114244031335228433922, 2862001783406012428802, 72067481493990612275474

Offset

0,2

Comments

In general, for k > 1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial(k*j,j) / ((k-1)*j+1) ~ (1-r)^(1/(k-1) - 1/2) * sqrt(1 + (k-1)*r) / (sqrt(2*Pi*(k-1)) * k^(1/(k-1) + 1/2) * n^(3/2) * r^(n + 1/(k-1))), where r is the smallest real root of the equation (k-1)^(k-1) * (1-r)^k = k^k * r. - Vaclav Kotesovec, Nov 15 2021

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

Formula

a(n) = Sum_{k=0..n} binomial(n+7*k,8*k) * binomial(8*k,k) / (7*k+1).

a(n) = F([(1+n)/7, (2+n)/7, (3+n)/7, (4+n)/7, (5+n)/7, (6+n)/7, 1+n/7, -n], [2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021

a(n) ~ sqrt(1 + 7*r) / (2^(17/7) * sqrt(7*Pi) * (1-r)^(5/14) * n^(3/2) * r^(n + 1/7)), where r = 0.036466941615119756839260438459647497790132092200414533994... is the smallest real root of the equation 7^7 * (1-r)^8 = 8^8 * r. - Vaclav Kotesovec, Nov 15 2021

Mathematica

nmax = 18; A[_] = 0; Do[A[x_] = (1 + x A[x]^8)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 7 k, 8 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]

Crossrefs

Cf. A006318, A007556, A346626, A346650, A349310, A349311, A349312, A349313.

Sequence in context: A351275 A364400 A334242 * A365555 A355472 A295183

Adjacent sequences: A349311 A349312 A349313 * A349315 A349316 A349317

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 14 2021

Status

approved