A118184 Column 0 of the matrix log of triangle A118180, after term in row n is multiplied by n: a(n) = n*[log(A118180)](n,0), where A118180(n,k) = 3^(k*(n-k)).

0, 1, -1, 3, -23, 329, 18231, -22030373, 34718491601, -130548608723439, 1300095260497408879, -35497483240662990289357, 2687397326811421691366217657, -562747611676887059779727492799911, 320110532506391993959111359699070808231

Offset: 0

Author

Paul D. Hanna, Apr 15 2006

Keywords

• sign

Comments

The entire matrix log of triangle A118180 is determined by column 0 (this sequence): [log(A118180)](n,k) = a(n-k)/(n-k)*(3^k)^(n-k) for n>k>=0.

Examples

			Column 0 of log(A118180) = [0, 1, -1/2, 3/3, -23/4, 329/5, 18231/6, ...].
The g.f. is illustrated by:
x/(1-x)^2 = x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + ...
= x/(1-3*x) - x^2/(1-9*x) + 3*x^3/(1-27*x) - 23*x^4/(1-81*x) + 329*x^5/(1-243*x) + 18231*x^6/(1-729*x) - 22030373*x^7/(1-2187*x) + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..65

Crossrefs

Cf. A118180, A118183.

Programs

Mathematica

A118183[n_]:= A118183[n]= If[n<2, (-1)^n, -Sum[3^(j*(n-j))*A118183[j], {j,0,n-1}]];
a[n_]:= a[n]= -Sum[3^(j*(n-j))*j*A118183[j], {j, 0, n}];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jun 29 2021 *)

PARI

{a(n)=local(T=matrix(n+1,n+1,r,c,if(r>=c,(3^(c-1))^(r-c))), L=sum(m=1,#T,-(T^0-T)^m/m));return(n*L[n+1,1])}

Sage

@CachedFunction
def A118183(n): return (-1)^n if (n<2) else -sum(3^(j*(n-j))*A118183(j) for j in (0..n-1))
def a(n): return (-1)*sum( 3^(j*(n-j))*j*A118183(j) for j in (0..n))
[a(n) for n in (0..30)] # G. C. Greubel, Jun 29 2021

Formula

G.f.: x/(1-x)^2 = Sum_{n>=0} a(n)*x^n/(1-3^n*x).

By using the inverse transformation: a(n) = Sum_{k=0..n} k*A118183(n-k)*(3^k)^(n-k) for n>=0.

a(3^n) is divisible by 3^n.