0,5

The entire matrix log of triangle A117401 is determined by column 0 (this sequence): [LOG(A117401)](n,k) = a(n-k)/(n-k)*(2^k)^(n-k) for n>k>=0.

G.f.: x/(1-x)^2 = Sum_{n>=0} a(n)*x^n/(1-2^n*x). By using the inverse transformation: a(n) = Sum_{k=0..n} k*A118196(n-k)*(2^k)^(n-k) for n>=0. a(2^n) is divisible by 2^n.

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

Column 0 of LOG(A117401) = [0, 1, 0, -1/3, 4/4, -11/5, -186/6,...] and

consists of terms a(n)/n (n>0); these terms are integers at n =

[0,1,2,4,6,8,10,14,16,22,26,32,34,38,46,50,58,62,64,70,...].

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/(1-2*x) - 0*x^2/(1-4*x) - 1*x^3/(1-8*x) + 4*x^4/(1-16*x) - 11*x^5/(1-32*x) - 186*x^6/(1-64*x) + 10823*x^7/(1-128*x) +...

Define g.f.: G(x) = Sum_{n>=1} a(n)*x^n/[n * 2^(n(n-1)/2)], then G(x) = x + 0x^2/4 - x^3/24 + 4x^4/256 - 11x^5/5120 - 186x^6/196608 +... and exp(G(x)) = 1 + x + x^2/2 + x^3/8 + x^4/64 + x^5/1024 + x^6/32768 +...

{a(n)=local(T=matrix(n+1, n+1, r, c, if(r>=c, (2^(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*(n-1)/2)*polcoeff(log(sum(k=0, n, x^k/2^(k*(k-1)/2))+x*O(x^n)), n)}

Cf. A117401, A118196.

Cf. A134531.

Sequence in context: A006248 A119571 A089920 this_sequence A092658 A113314 A014458

Adjacent sequences: A118194 A118195 A118196 this_sequence A118198 A118199 A118200

sign

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 15 2006, Oct 30 2007