A346668 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(8*k,k) / (7*k + 1).

1, 0, 7, 70, 917, 12922, 192591, 2984156, 47594289, 776184997, 12884436285, 216981375849, 3698021707457, 63663537870121, 1105474964523293, 19339098305850757, 340519405008643561, 6030158137055187758, 107328892461895007043, 1918980244360791943044, 34450128513971163342013

Offset

0,3

Comments

Inverse binomial transform of A007556.

In general, for m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(m*k,k) / ((m-1)*k + 1) ~ (m^m - (m-1)^(m-1))^(n + 3/2) / (sqrt(2*Pi) * m^((3*m-1)/2) * n^(3/2) * (m-1)^((m-1)*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

Links

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

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^6 * A(x)^8.

G.f.: Sum_{k>=0} ( binomial(8*k,k) / (7*k + 1) ) * x^k / (1 + x)^(k+1).

a(n) ~ 15953673^(n + 3/2) / (34359738368 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

Mathematica

Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[_] = 0; Do[A[x_] = 1/(1 + x) + x (1 + x)^6 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 20; CoefficientList[Series[Sum[(Binomial[8 k, k]/(7 k + 1)) x^k/(1 + x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[(-1)^n HypergeometricPFQ[{1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, -n}, {2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7}, 16777216/823543], {n, 0, 20}]

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*binomial(8*k, k)/(7*k + 1)); \\ Michel Marcus, Jul 28 2021

Crossrefs

Cf. A005043, A007556, A346628, A346650, A346664, A346665, A346666, A346667.

Sequence in context: A141151 A001669 A051604 * A362775 A365031 A097630

Adjacent sequences: A346665 A346666 A346667 * A346669 A346670 A346671

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 27 2021

Status

approved