cp's OEIS Frontend

A101617 The trinomial transform (A027907) gives powers of 3, while the trinomial transform of this sequence shift one place left gives powers of 5.

%I A101617 #12 Apr 18 2020 00:02:17
%S A101617 1,1,1,3,-3,19,-43,139,-355,995,-2587,6907,-17939,46931,-121419,
%T A101617 314603,-811203,2091459,-5379963,13833179,-35527795,91210035,
%U A101617 -234020267,600258507,-1539135779,3945762211,-10113490139,25918908603,-66417608403,170182721299,-436032111883,1117120911019
%N A101617 The trinomial transform (A027907) gives powers of 3, while the trinomial transform of this sequence shift one place left gives powers of 5.
%F A101617 G.f.: A(x) = (1 + 3*x - 2*x^2 - 6*x^3)/(1 + 2*x - 5*x^2 - 6*x^3 + 8*x^4). [corrected by _Georg Fischer_, Apr 17 2020]
%F A101617 3^n = Sum_{k=0..2*n} A027907(n, k)*a(k) for n>=0 and
%F A101617 5^n = Sum_{k=0..2*n} A027907(n, k)*a(k+1) for n>=0.
%F A101617 a(n) = (-1)^n*A006131(n-1) + (1/3)[(-2)^n + 2]. - _Ralf Stephan_, May 16 2007
%e A101617 3^3 = 1*(1) + 3*(1) + 6*(1) + 7*(3) + 6*(-3) + 3*(19) + 1*(-43).
%e A101617 5^3 = 1*(1) + 3*(1) + 6*(3) + 7*(-3) + 6*(19) + 3*(-43) + 1*(139).
%e A101617 In general, a sequence A with the property that the
%e A101617 trinomial transform of A gives powers of P, while the
%e A101617 trinomial transform of LSHIFT(A) gives powers of Q
%e A101617 has the g.f.: N(x)/D(x) where
%e A101617 N(x)=(1+3*x-(Q-3)*x^2-(P+Q-2)*x^3) and
%e A101617 D(x)=(1+2*x-(P+Q-3)*x^2-(P+Q-2)*x^3+(P-1)*(Q-1)*x^4).
%t A101617 nn:=31; CoefficientList[Series[(1 + 3*x - 2*x^2 - 6*x^3)/(1 + 2*x - 5*x
%t A101617 ^2 - 6*x^3 + 8*x^4),{x,0,nn}],x] (* _Georg Fischer_, Apr 17 2020 *)
%o A101617 (PARI) {a(n)=local(P=3,Q=5,V=[1,1]);if(n>1, for(m=1,n, V=concat(V,P^m-sum(k=0,2*m-1,polcoeff((1+x+x^2)^m+x*O(x^k),k)*V[k+1])); V=concat(V,Q^m-sum(k=0,2*m-1,polcoeff((1+x+x^2)^m+x*O(x^k),k)*V[k+2])); ));V[n+1]}
%o A101617 for(n=0,30,print1(a(n),", "))
%Y A101617 Cf. A027907, A100321.
%K A101617 sign
%O A101617 0,4
%A A101617 _Paul D. Hanna_, Dec 09 2004