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
Keywords
Examples
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)
Links
- G. C. Greubel, Table of n, a(n) for n = 0..55
Programs
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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, 15}] (* G. C. Greubel, Jun 29 2021 *)
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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
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
Comments