A193016 - OEIS

%I #11 Jan 18 2022 02:36:03

%S 1,3,7,7,6,7,5,3,2,7,4,9,0,9,4,6,5,4,6,2,1,1,5,6,5,1,2,1,0,7,0,3,9,1,

%T 7,7,3,6,9,5,8,3,5,1,5,6,0,4,1,3,1,2,2,0,0,2,6,7,3,2,1,5,9,2,5,7,6,0,

%U 2,5,7,9,2,0,9,9,3,9,1,1,3,0,2,1,8,1,0,8,8,7,7,0,5,3,3,3,0,5,3,6

%N Decimal expansion of the coefficient of x in the reduction of sinh(x) by x^2->x+1.

%C Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.

%F From _Amiram Eldar_, Jan 18 2022: (Start)

%F Equals Sum_{k>=0} Fibonacci(2*k+1)/(2*k+1)!.

%F Equals (e+1)*sinh(sqrt(5)/2)/sqrt(5*e). (End)

%e 1.3776753274909465462115651210703917736958351560...

%t f[x_] := Sinh[x]; r[n_] := Fibonacci[n];

%t c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]

%t u1 = N[Sum[c[n]*r[n], {n, 0, 300}], 100]

%t RealDigits[u1, 10]

%Y Cf. A000045, A193010, A192232, A193015.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Jul 14 2011