cp's OEIS Frontend

A090158 Odd-indexed terms of the binomial transform equals 1 and the even-indexed terms of the second binomial transform equals 1.

%I A090158 #9 Feb 13 2016 09:47:28
%S A090158 1,0,-3,9,-15,15,-63,399,-255,-7425,-1023,355839,-4095,-22360065,
%T A090158 -16383,1903790079,-65535,-209865211905,-262143,29088885637119,
%U A090158 -1048575,-4951498051026945,-4194303,1015423886515240959,-16777215,-246921480190174429185
%N A090158 Odd-indexed terms of the binomial transform equals 1 and the even-indexed terms of the second binomial transform equals 1.
%C A090158 Compare the first and 2nd binomial transforms of this sequence:
%C A090158 first binomial={1,1,-2,1,4,1,-62,1,1384,1,-50522,1,2702764,..};
%C A090158 2nd binomial={1,2,1,-1,1,17,1,-271,1,7937,1,-353791,..};
%C A090158 to that of the first and 2nd binomial transforms of A090145:
%C A090158 first binomial of A090145={1,0,1,-3,1,15,1,-273,1,7935,1,..};
%C A090158 2nd binomial of A090145={1,1,2,1,-4,1,62,1,-1384,1,50522,..}.
%C A090158 Comparison reveals this e.g.f. relation of the two sequences:
%C A090158 e.g.f.: exp(x)*G090158(x) + exp(2x)*G090145(x) = 2 + 2*sinh(x);
%C A090158 e.g.f.: exp(2*x)*G090158(x) - exp(x)*G090145(x) = 2*sinh(x);
%C A090158 thus G090158(x) = 2*(1+sinh(x) + exp(x)*sinh(x))/(exp(x)*(1+exp(2*x)))
%C A090158 G090145(x) = 2*((1+sinh(x))*exp(x) - sinh(x))/(exp(x)*(1+exp(2*x))).
%F A090158 E.g.f.: 2*(1 + sinh(x) + exp(x)*sinh(x)) / (exp(x)*(1 + exp(2*x))).
%F A090158 a(2n) = 1 - 2^(2n);
%F A090158 1 = sum_{k=0..2n-1} C(2n-1, k)*a(k);
%F A090158 1 = sum_{k=0..2n} 2^(2n-k)*C(2n, k)*a(k).
%t A090158 With[{nn=30},CoefficientList[Series[2 (1+Sinh[x]+Exp[x]Sinh[x])/ (Exp[x] (1+ Exp[2x])),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Feb 13 2016 *)
%Y A090158 Cf. A090145.
%K A090158 sign
%O A090158 0,3
%A A090158 _Paul D. Hanna_, Nov 22 2003