A118189 Column 0 of the matrix log of triangle A118185, after term in row n is multiplied by n: a(n) = n*[log(A118185)](n,0), where A118185(n,k) = 4^(k*(n-k)).

0, 1, -2, 19, -764, 125701, -83499002, 222705979399, -2379643407695864, 101770765968904486921, -17414214749792087566712822, 11920352399707142353576549941259, -32640155138015817553201240150152052724, 357505372216293786145503061380504161718632461

Offset

0,3

Comments

The entire matrix log of triangle A118185 is determined by column 0 (this sequence): [log(A118185)](n,k) = a(n-k)*4^(k*(n-k))/(n-k) for n > k >= 0.

Links

G. C. Greubel, Table of n, a(n) for n = 0..55

Formula

G.f.: x/(1-x)^2 = Sum_{n>=0} a(n)*x^n/(1-4^n*x).

By using the inverse transformation: a(n) = Sum_{k=0..n} k*A118188(n-k)*4^(k*(n-k)) for n>=0.

a(2^n) is divisible by 2^n.

L.g.f.: Sum_{n>=1} a(n)*x^n/[n*2^(n^2)] = log( Sum_{n>=0} x^n/2^(n^2) ). - Paul D. Hanna, Oct 14 2009

Example

Column 0 of log(A118185) = [0, 1, -2/2, 19/3, -764/4, 125701/5, ...].
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/(1-4*x) - 2*x^2/(1-16*x) + 19*x^3/(1-64*x) - 764*x^4/(1-256*x) + 125701*x^5/(1-1024*x) - 83499002*x^6/(1-4096*x) + 222705979399*x^7/(1-16384*x) + ...
From Paul D. Hanna, Oct 14 2009: (Start)
Illustrate the logarithmic g.f. by:
L(x) = x/2^1 - 2*x^2/(2*2^4) + 19*x^3/(3*2^9) - 764*x^4/(4*2^16) +- ...
where exp(L(x)) = 1 + x/2^1 + x^2/2^4 + x^3/2^9 + x^4/2^16 + ... (End)

Mathematica

A118188[n_]:= A118188[n]= If[n<2, (-1)^n, -Sum[4^(j*(n-j))*A118188[j], {j, 0, n-1}]];
a[n_]{= a[n]= -Sum[4^(j*(n-j))*j*A118188[j], {j, 0, n}];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jun 29 2021 *)

Prog

(PARI) {a(n)=local(T=matrix(n+1, n+1, r, c, if(r>=c, (4^(c-1))^(r-c))), L=sum(m=1, #T, -(T^0-T)^m/m)); return(n*L[n+1, 1])}
(PARI) {a(n)=n*2^(n^2)*polcoeff(log(sum(m=0, n, x^m/2^(m^2))+x*O(x^n)), n)} \\ Paul D. Hanna, Oct 14 2009
(Sage)
@CachedFunction
def A118188(n): return (-1)^n if (n<2) else -sum(4^(j*(n-j))*A118188(j) for j in (0..n-1))
def a(n): return (-1)*sum(4^(j*(n-j))*j*A118188(j) for j in (0..n))
[a(n) for n in (0..30)] # G. C. Greubel, Jun 29 2021

Crossrefs

Cf. A118185 (triangle), A118188.

Sequence in context: A278840 A300995 A086976 * A062623 A155956 A013047

Adjacent sequences: A118186 A118187 A118188 * A118190 A118191 A118192

Keyword

sign

Author

Paul D. Hanna, Apr 15 2006

Status

approved