%I #29 Nov 13 2024 22:31:50

%S 1,1,7,161,7631,607009,72605303,12172272321,2722634203807,

%T 783282749905601,281751782666559239,123890976070562785633,

%U 65380371270827869603439,40779819387085820255904481,29677003954344675666092048791,24921035407468294238607282809729

%N a(n) = A271697(2*n, n).

%C Central coefficients of the triangles A046739 and A271697.

%H Andrew Howroyd, <a href="/A320337/b320337.txt">Table of n, a(n) for n = 0..100</a>

%F 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

%F a(n) ~ sqrt(3) * 2^(2*n + 1) * n^(2*n) / exp(2*n + 1). - _Vaclav Kotesovec_, Dec 19 2018

%p 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

%t 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];

%t a[n_] := Sum[(-1)^(n - k) E1[n + k, n] Binomial[2 n, n - k], {k, 0, n}];

%t Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Dec 30 2018, after _Peter Luschny_ *)

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

%K nonn

%O 0,3

%A _Maxwell Jiang_, Dec 18 2018 (added without permission by editors)