A004254 -id:A004254 - cp's OEIS frontend

A179900 Triangle T(n,k) read by rows: coefficient of [x^k] of the polynomial p_n(x)=(5-x)*p_{n-1}(x)-p_{n-2}(x), p_0=1, p_1=5-x.

1, 5, -1, 24, -10, 1, 115, -73, 15, -1, 551, -470, 147, -20, 1, 2640, -2828, 1190, -246, 25, -1, 12649, -16310, 8631, -2400, 370, -30, 1, 60605, -91371, 58275, -20385, 4225, -519, 35, -1, 290376, -501150, 374115, -157800, 41140, -6790, 693, -40, 1, 1391275

Offset: 0

Roger L. Bagula, Jul 31 2010

The row sums are 1, 4, 15, 56, 209, 780, 2911, .. A001353.
Apart from signs, the same as A123967.
This can also be defined as the coefficients of the characteristic polynomial of the n X n tridiagonal symmetric matrix with 5's on the diagonal and -1's on the two adjacent subdiagonals. Expansion of the determinant along the first column yields the recurrence of the definition.

			1 ;       # 1
5, -1;     # 5-x
24, -10, 1 ;  # 24-10x+x^2
115, -73, 15, -1; # 115-73x+15x^2-x^3
551, -470, 147, -20, 1;
2640, -2828, 1190, -246, 25, -1;
12649, -16310, 8631, -2400, 370, -30, 1;
60605, -91371, 58275, -20385, 4225, -519, 35, -1;
290376, -501150, 374115, -157800, 41140, -6790, 693, -40, 1;
1391275, -2704755, 2313450, -1142730, 359275, -74571, 10220, -892, 45, -1;

Clear[M, T, d, a, x, a0]
T[n_, m_, d_] := If[ n == m, 5, If[n == m - 1 || n == m + 1, -1, 0]]
M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]
Table[Det[M[d]], {d, 1, 10}]
Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]
a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], { d, 1, 10}]]
Flatten[a]
MatrixForm[a]

T(n,k) = 5*T(n-1,k)-T(n-1,k-1)-T(n-2,k) starting T(0,0)=1, T(1,0)=5 and T(1,1)=-1.

T(n,0) = A004254(n+1).

A362357 Bisection of Chebyshev {S(n, 5)}_{n>=0}; the even part.

Original entry on oeis.org

1, 24, 551, 12649, 290376, 6665999, 153027601, 3512968824, 80645255351, 1851327904249, 42499896542376, 975646292570399, 22397364832576801, 514163744856696024, 11803368766871431751, 270963317893186234249

Offset: 0

Wolfdieter Lang, Apr 26 2023

The odd part of this bisection is given by 5*A097778(n), for n >= 0.

Index entries for sequences related to Chebyshev polynomials.`
Index entries for linear recurrences with constant coefficients, signature (23,-1).

Cf. A004254, A049310, A097778.

Table[ChebyshevU[2*n, 5/2], {n, 0, 20}] (* Vaclav Kotesovec, May 27 2023 *)

a(n) = polchebyshev(2*n, 2, 5/2); \\ Michel Marcus, May 27 2023

a(n) = S(2*n, 5) = S(n, 23) + S(n-1, 23), with the Chebyshev S polynomials (see A049310), S(-1, x) = 0, S(n, 5) = A004254(n+1) and S(n, 23) = A097778(n).
O.g.f.: (1 + x)/(1 - 23*x + x^2).
a(n) = 23*a(n-1) - a(n-2), for n >= 0, with a(-1) = -1 and a(-2) = -24.

cp's OEIS Frontend

A179900 Triangle T(n,k) read by rows: coefficient of [x^k] of the polynomial p_n(x)=(5-x)*p_{n-1}(x)-p_{n-2}(x), p_0=1, p_1=5-x.

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