A320337 a(n) = A271697(2*n, n).

1, 1, 7, 161, 7631, 607009, 72605303, 12172272321, 2722634203807, 783282749905601, 281751782666559239, 123890976070562785633, 65380371270827869603439, 40779819387085820255904481, 29677003954344675666092048791, 24921035407468294238607282809729

Offset: 0

Author

Maxwell Jiang, Dec 18 2018 (added without permission by editors)

Keywords

• nonn

Comments

Central coefficients of the triangles A046739 and A271697.

Links

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

Crossrefs

Cf. A173018, A046739, A062868, A180056, A271697.

Programs

Maple

a := n -> add((-1)^(n-k)*combinat:-eulerian1(n+k,n)*binomial(2*n,n-k), k=0..n): seq(a(n), n=0..15); # Peter Luschny, Dec 19 2018

Mathematica

E1[n_ /; n >= 0, 0] = 1; E1[n_, k_] /; k < 0 || k > n = 0; E1[n_, k_] := E1[n, k] = (n - k) E1[n - 1, k - 1] + (k + 1) E1[n - 1, k];
a[n_] := Sum[(-1)^(n - k) E1[n + k, n] Binomial[2 n, n - k], {k, 0, n}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Dec 30 2018, after Peter Luschny *)

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*E(n+k, n)*binomial(2*n,n-k) where E are the Eulerian numbers A173018. - Peter Luschny, Dec 19 2018

a(n) ~ sqrt(3) * 2^(2*n + 1) * n^(2*n) / exp(2*n + 1). - Vaclav Kotesovec, Dec 19 2018