A101617 The trinomial transform (A027907) gives powers of 3, while the trinomial transform of this sequence shift one place left gives powers of 5.
1, 1, 1, 3, -3, 19, -43, 139, -355, 995, -2587, 6907, -17939, 46931, -121419, 314603, -811203, 2091459, -5379963, 13833179, -35527795, 91210035, -234020267, 600258507, -1539135779, 3945762211, -10113490139, 25918908603, -66417608403, 170182721299, -436032111883, 1117120911019
Offset: 0
Keywords
Examples
3^3 = 1*(1) + 3*(1) + 6*(1) + 7*(3) + 6*(-3) + 3*(19) + 1*(-43). 5^3 = 1*(1) + 3*(1) + 6*(3) + 7*(-3) + 6*(19) + 3*(-43) + 1*(139). In general, a sequence A with the property that the trinomial transform of A gives powers of P, while the trinomial transform of LSHIFT(A) gives powers of Q has the g.f.: N(x)/D(x) where N(x)=(1+3*x-(Q-3)*x^2-(P+Q-2)*x^3) and D(x)=(1+2*x-(P+Q-3)*x^2-(P+Q-2)*x^3+(P-1)*(Q-1)*x^4).
Programs
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Mathematica
nn:=31; CoefficientList[Series[(1 + 3*x - 2*x^2 - 6*x^3)/(1 + 2*x - 5*x ^2 - 6*x^3 + 8*x^4),{x,0,nn}],x] (* Georg Fischer, Apr 17 2020 *) -
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]} for(n=0,30,print1(a(n),", "))
Formula
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]
3^n = Sum_{k=0..2*n} A027907(n, k)*a(k) for n>=0 and
5^n = Sum_{k=0..2*n} A027907(n, k)*a(k+1) for n>=0.
a(n) = (-1)^n*A006131(n-1) + (1/3)[(-2)^n + 2]. - Ralf Stephan, May 16 2007