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A118189
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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).
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1
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0, 1, -2, 19, -764, 125701, -83499002, 222705979399, -2379643407695864, 101770765968904486921, -17414214749792087566712822, 11920352399707142353576549941259, -32640155138015817553201240150152052724, 357505372216293786145503061380504161718632461
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OFFSET
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0,3
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COMMENTS
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The entire matrix log of triangle A118185 is determined by column 0 (this sequence): [log(A118185)](n,k) = a(n-k)/(n-k)*(4^k)^(n-k) for n>k>=0.
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LINKS
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Table of n, a(n) for n=0..13.
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FORMULA
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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) ). [From Paul D. Hanna, Oct 14 2009]
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EXAMPLE
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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) +...
Contribution 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)
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PROG
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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)} [From Paul D. Hanna, Oct 14 2009]
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CROSSREFS
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Cf. A118185 (triangle), A118188.
Sequence in context: A158099 A015204 A086976 * A062623 A155956 A013047
Adjacent sequences: A118186 A118187 A118188 * A118190 A118191 A118192
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KEYWORD
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sign
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AUTHOR
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Paul D. Hanna, Apr 15 2006
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STATUS
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approved
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