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A373504 Triangular array: row n gives the coefficients T(n,k) of powers x^(2k) in the series expansion of ((b^n + b^(-n))/2)^2, where b = x + sqrt(x^2 + 1).

%I A373504 #37 Aug 11 2024 14:17:36
%S A373504 1,1,1,1,4,4,1,9,24,16,1,16,80,128,64,1,25,200,560,640,256,1,36,420,
%T A373504 1792,3456,3072,1024,1,49,784,4704,13440,19712,14336,4096,1,64,1344,
%U A373504 10752,42240,90112,106496,65536,16384,1,81,2160,22176,114048,329472,559104,552960,294912,65536
%N A373504 Triangular array: row n gives the coefficients T(n,k) of powers x^(2k) in the series expansion of ((b^n + b^(-n))/2)^2, where b = x + sqrt(x^2 + 1).
%C A373504 Related to Chebyshev polynomials of the first kind; see A123588.
%F A373504 T(n, k) = if (k=0) then 1, otherwise 4^(k - 1)*(2*binomial(n + k, 2*k) - binomial(n + k - 1, 2*k - 1)). - _Detlef Meya_, Aug 11 2024
%e A373504 First 8 rows:
%e A373504   1
%e A373504   1    1
%e A373504   1    4     4
%e A373504   1    9    24     16
%e A373504   1   16    80    128     64
%e A373504   1   25   200    560    640    256
%e A373504   1   36   420   1792   3456   3072   1024
%e A373504   1   49   784   4704  13440  19612  14336  4096
%e A373504 The 4th polynomial is 1 + 9 x^2 + 24 x^4 + 16 x^6.
%p A373504 p:= proc(n) option remember; (b-> series(
%p A373504       ((b^n+b^(-n))/2)^2, x, 2*n+1))(x+sqrt(x^2+1))
%p A373504     end:
%p A373504 T:= (n, k)-> coeff(p(n), x, 2*k):
%p A373504 seq(seq(T(n,k), k=0..n), n=0..10);  # _Alois P. Heinz_, Aug 03 2024
%t A373504 t[n_] := ((x + Sqrt[x^2 + 1])^n + (x + Sqrt[x^2 + 1])^(-n))/2
%t A373504 u = Expand[Table[FullSimplify[Expand[t[n]]], {n, 0, 10}]^2]
%t A373504 v = Column[CoefficientList[u, x^2]] (* array *)
%t A373504 Flatten[v] (* sequence *)
%t A373504 T[n_, k_] := If[k==0, 1, 4^(k - 1)*(2*Binomial[n + k, 2*k] - Binomial[n + k -1, 2*k -1])]; Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* _Detlef Meya_, Aug 11 2024 *)
%Y A373504 Cf. A000012 (col 0), A000290 (col 1), A002415 ((1/4)*col(2)), A112742 (col 2), A000302 (T(n,n)), A123588, A008310.
%Y A373504 Row sums give A055997(n+1).
%Y A373504 Triangle without column 0 gives A334009.
%K A373504 nonn,tabl
%O A373504 0,5
%A A373504 _Clark Kimberling_, Aug 03 2024