A396724 - OEIS

A396724

a(n) = (-4^n/2) p(n, -3/2), where p(n,x) is the Lagrange interpolating polynomial through (1,0) and (k,1) for k=2..n+1.

%I #14 Jun 30 2026 07:23:30

%S 0,5,55,430,2875,17506,100054,546076,2877139,14741786,73840002,

%T 362963908,1756073182,8381571028,39537088012,184595889208,

%U 854091531907,3920036136778,17862714586762,80870371149748,363985389650602,1629529723109308,7259813777148532

%N a(n) = (-4^n/2) p(n, -3/2), where p(n,x) is the Lagrange interpolating polynomial through (1,0) and (k,1) for k=2..n+1.

%F Conjecture D-finite with recurrence n*a(n) +2*(-4*n-3)*a(n-1) +8*(2*n+3)*a(n-2)=0. - _R. J. Mathar_, Jun 30 2026

%t t[0] = {0}; t[n_] := t[n] = Join[t[n - 1], {1}];

%t p[n_, x_] := Expand[InterpolatingPolynomial[t[n], x]];

%t Table[(-4^n/2) p[n, -3/2], {n, 0, 40}]

%t Labeled[Plot[{p[1, x], p[2, x], p[3, x], p[4, x], p[5, x]}, {x, -1, 7}], "Plot of p[n,x] for n=1..5."]

%Y Cf. A395640, A004442, A125577, A000531, A000346, A396722, A396723.

%K nonn,changed

%O 0,2

%A _Clark Kimberling_, Jun 20 2026