A166357 -id:A166357 - cp's OEIS frontend

A166359 Diagonal sums of the exponential Riordan array [1+x*arctanh(x), x], A166357.

1, 3, 15, 197, 6909, 459383, 48252699, 7299708105, 1499523879481, 401147660278507, 135421121289695655, 56285769483090611085, 28237152577765405343285, 16821061018350834178553055

Offset: 0

Paul Barry, Oct 12 2009

Aeration of this sequence gives the diagonal sums of A166357.

Andrew Howroyd, Table of n, a(n) for n = 0..100

Cf. A166356, A166357.

Table[1 + Sum[Binomial[n + k, 2*k] * ((2*k-1)! + (2*k-2)!), {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 17 2018 *)

a(n)={1 + sum(k=1, n, binomial(n + k, 2*k) * ((2*k-1)! + (2*k-2)!))} \\ Andrew Howroyd, Aug 17 2018

a(n) = Sum_{k=0..n} C(n+k,2k)*A166356(k).

a(n) ~ sqrt(Pi) * 2^(2*n) * n^(2*n - 1/2) / exp(2*n). - Vaclav Kotesovec, Aug 17 2018

A166358 Row sums of exponential Riordan array [1+x*arctanh(x), x], A166357.

Original entry on oeis.org

1, 1, 3, 7, 21, 61, 295, 1331, 10409, 65017, 694411, 5454879, 73145149, 689074101, 11090013103, 121652191051, 2282132463953, 28550033871857, 611369381873683, 8587415858721079, 206626962757626981, 3219065122124476717

Offset: 0

Paul Barry, Oct 12 2009

Binomial transform of aeration of A166356.

Cf. A166356, A166357.

(* The function RiordanArray is defined in A256893. *)
nmax = 21; R = RiordanArray[1 + # ArcTanh[#]&, #&, nmax + 1, True];
Total /@ R (* Jean-François Alcover, Jul 20 2019 *)

E.g.f.: exp(x)*(1+x*arctanh(x)).
a(n) = Sum_{k=0..n} C(n,k)*A166356(k/2)*(1+(-1)^k)/2.
a(n) ~ (exp(1) + (-1)^n*exp(-1)) * (n-1)! / 2. - Vaclav Kotesovec, Aug 17 2018

cp's OEIS Frontend

A166359 Diagonal sums of the exponential Riordan array [1+x*arctanh(x), x], A166357.

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