A342168 - OEIS

%I #32 Jun 28 2024 01:20:13

%S 1,4,24,204,2255,30744,499121,9409960,202176360,4878316860,

%T 130651068911,3846719565780,123517560398401,4296240885694576,

%U 160935647131239840,6460088606857290384,276655979838719058119,12591439417867717440180,606947064800948702246681

%N a(n) = U(n, (n+3)/2) where U(n, x) is a Chebyshev polynomial of the 2nd kind.

%H Seiichi Manyama, <a href="/A342168/b342168.txt">Table of n, a(n) for n = 0..386</a>

%H Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, <a href="https://arxiv.org/abs/2406.12941">Metered Parking Functions</a>, arXiv:2406.12941 [math.CO], 2024. See pp. 11, 22.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.

%F a(n) = Sum_{k=0..n} (n+1)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (n+1)^k * binomial(n+1+k,2*k+1).

%F a(n) ~ exp(3) * n^n. - _Vaclav Kotesovec_, May 06 2021

%t Table[ChebyshevU[n, (n + 3)/2], {n, 0, 18}] (* _Amiram Eldar_, Apr 27 2021 *)

%o (PARI) a(n) = polchebyshev(n, 2, (n+3)/2);

%o (PARI) a(n) = sum(k=0, n, (n+1)^(n-k)*binomial(2*n+1-k, k));

%o (PARI) a(n) = sum(k=0, n, (n+1)^k*binomial(n+1+k, 2*k+1));

%Y Cf. A097690, A097691, A323118, A342167.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 03 2021