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A118183 Column 0 of the matrix inverse of triangle A118180. 4
1, -1, 2, -10, 134, -4942, 505682, -142838074, 108933186230, -210663798566302, 812745803173573538, 6022271614633142122646, -2489044042602910169970590746, 996768343710992528631250678460690, -928936693384587466168289179772677376782 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The entire matrix inverse of triangle A118180 is determined by column 0 (this sequence): [A118180^-1](n,k) = a(n-k)*(3^k)^(n-k) for n>=k>=0. Any g.f. of the form: Sum_{k>=0} b(k)*x^k may be expressed as: Sum_{n>=0} c(n)*x^n/(1-3^n*x) by applying the inverse transformation: c(n) = Sum_{k=0..n} a(n-k)*b(k)*(3^k)^(n-k).

LINKS

Table of n, a(n) for n=0..14.

FORMULA

G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1-3^n*x). 0^n = Sum_{k=0..n} a(k)*(3^k)^(n-k) for n>=0.

EXAMPLE

Recurrence at n=4:

0 = a(0)*(3^0)^4 +a(1)*(3^1)^3 +a(2)*(3^2)^2 +a(3)*(3^3)^1 +a(4)*(3^4)^0

= 1*(3^0) - 1*(3^3) + 2*(3^4) - 10*(3^3) + 134*(3^0).

The g.f. is illustrated by:

1 = 1/(1-x) -1*x/(1-3*x) +2*x^2/(1-9*x) -10*x^3/(1-27*x) +134*x^4/(1-81*x)

- 4942*x^5/(1-243*x) +505682*x^6/(1-729*x) -142838074*x^7/(1-2187*x) +...

PROG

(PARI) {a(n)=local(T=matrix(n+1, n+1, r, c, if(r>=c, (3^(c-1))^(r-c)))); return((T^-1)[n+1, 1])}

CROSSREFS

Cf. A118180.

Sequence in context: A254431 A011838 A336537 * A134051 A075199 A134981

Adjacent sequences:  A118180 A118181 A118182 * A118184 A118185 A118186

KEYWORD

sign

AUTHOR

Paul D. Hanna, Apr 15 2006

STATUS

approved

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Last modified August 11 15:12 EDT 2020. Contains 336428 sequences. (Running on oeis4.)