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A101617
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The trinomial transform (A027907) gives powers of 3, while the trinomial transform of this sequence shift one place left gives powers of 5.
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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FORMULA
| G.f.: A(x) = (1 + 3*x - 2*x^2)/(1 + 2*x - 5*x^2 - 6*x^3 + 8*x^4). 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
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EXAMPLE
| 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).
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PROG
| (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]}
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CROSSREFS
| Cf. A027907, A100321.
Sequence in context: A136475 A143180 A074249 * A131443 A157050 A059368
Adjacent sequences: A101614 A101615 A101616 * A101618 A101619 A101620
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KEYWORD
| sign
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Dec 09 2004
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