A342168
a(n) = U(n, (n+3)/2) where U(n, x) is a Chebyshev polynomial of the 2nd kind.
Original entry on oeis.org
1, 4, 24, 204, 2255, 30744, 499121, 9409960, 202176360, 4878316860, 130651068911, 3846719565780, 123517560398401, 4296240885694576, 160935647131239840, 6460088606857290384, 276655979838719058119, 12591439417867717440180, 606947064800948702246681
Offset: 0
- Seiichi Manyama, Table of n, a(n) for n = 0..386
- Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 11, 22.
- Wikipedia, Chebyshev polynomials.
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Table[ChebyshevU[n, (n + 3)/2], {n, 0, 18}] (* Amiram Eldar, Apr 27 2021 *)
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a(n) = polchebyshev(n, 2, (n+3)/2);
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a(n) = sum(k=0, n, (n+1)^(n-k)*binomial(2*n+1-k, k));
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a(n) = sum(k=0, n, (n+1)^k*binomial(n+1+k, 2*k+1));
A107995
Chebyshev polynomial of the second kind U[n,x] evaluated at x=n+2.
Original entry on oeis.org
1, 6, 63, 980, 20305, 526890, 16451071, 600940872, 25154396001, 1187422368110, 62418042417599, 3616337930622300, 228977061309711793, 15731733543660288210, 1165677769357309014015, 92665403695822344828176
Offset: 0
a(3)=980 because U[3,x]=8x^3-4x and U[3,5]=8*5^3-4*5=980.
- Rosenblum and Rovnyak, Hardy Classes and Operator Theory,Dover, New York,1985, page 18.
- G. Szego, Orthogonal polynomials, Amer. Math. Soc., Providence, 1939, p. 29.
- G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford, 1966, p. 35.
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with(orthopoly): seq(U(n,n+2),n=0..17);
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Table[ChebyshevU[n, n + 2], {n, 0, 15}] (* Amiram Eldar, Mar 05 2021 *)
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a(n) = polchebyshev(n, 2, n+2); \\ Seiichi Manyama, Mar 05 2021
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a(n) = sum(k=0, n, (2*n+2)^k*binomial(n+1+k, 2*k+1)); \\ Seiichi Manyama, Mar 05 2021
A343261
a(n) = 2 * T(n,(n+2)/2) where T(n,x) is a Chebyshev polynomial of the first kind.
Original entry on oeis.org
2, 3, 14, 110, 1154, 15127, 238142, 4379769, 92198402, 2186871698, 57721023502, 1678243366813, 53301709843202, 1836220544383695, 68200709735854334, 2716906424134261502, 115561578124838522882, 5227260815326346060059, 250566480717349417632398
Offset: 0
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Table[2*ChebyshevT[n, (n+2)/2], {n, 0, 18}] (* Amiram Eldar, Apr 09 2021 *)
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a(n) = 2*polchebyshev(n, 1, (n+2)/2);
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a(n) = round(2*cos(n*acos((n+2)/2)));
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a(n) = if(n==0, 2, 2*n*sum(k=0, n, n^k*binomial(n+k, 2*k)/(n+k)));
A372817
Table read by antidiagonals: T(m,n) = number of 1-metered (m,n)-parking functions.
Original entry on oeis.org
1, 0, 2, 0, 3, 3, 0, 4, 8, 4, 0, 6, 21, 15, 5, 0, 8, 55, 56, 24, 6, 0, 12, 145, 209, 115, 35, 7, 0, 16, 380, 780, 551, 204, 48, 8, 0, 24, 1000, 2912, 2640, 1189, 329, 63, 9, 0, 32, 2625, 10868, 12649, 6930, 2255, 496, 80, 10, 0, 48, 6900, 40569, 60606, 40391, 15456, 3905, 711, 99, 11
Offset: 1
For T(3,2) the 1-metered (3,2)-parking functions are 111, 121, 211, 212.
Table begins:
1, 2, 3, 4, 5, 6, 7, ...
0, 3, 8, 15, 24, 35, 48, ...
0, 4, 21, 56, 115, 204, 329, ...
0, 6, 55, 209, 551, 1189, 2255, ...
0, 8, 145, 780, 2640, 6930, 15456, ...
0, 12, 380, 2912, 12649, 40391, 105937, ...
0, 16, 1000, 10868, 60606, 235416, 726103, ...
...
- Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024.
Showing 1-4 of 4 results.