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A156101
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L.g.f.: A(x) = Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} (1 + 2^n*x)^n*x^n/n .
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1
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1, 5, 13, 65, 401, 3521, 43457, 738305, 17746177, 593695745, 27878501377, 1840450134017, 169904883945473, 22139372291866625, 4036405254299041793, 1038968242677362458625, 375102612647535161966593
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OFFSET
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1,2
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COMMENTS
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Compare to l.g.f. Sum_{m>=1} (1 + x)^m * x^m/m of the Fibonacci sequence.
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LINKS
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FORMULA
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L.g.f.: A(x) = log(G(x)) where G(x) is the g.f. of A156100.
a(n) = n*Sum_{k=0..floor(n/2)} C(n-k,k)*2^(k(n-k))/(n-k). - Paul D. Hanna, Apr 10 2009
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EXAMPLE
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G.f.: A(x) = x + 5*x^2/2 + 13*x^3/3 + 65*x^4/4 + 401*x^5/5 + ...
A(x) = (1 + 2*x)*x + (1 + 2^2*x)^2*x^2/2 + (1 + 2^3*x)^3*x^3/3 + ...
exp(A(x)) = 1 + x + 3*x^2 + 7*x^3 + 25*x^4 + 113*x^5 + 741*x^6 + ...
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MATHEMATICA
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Table[n*Sum[Binomial[n-k, k]*2^(k(n-k))/(n-k), {k, 0, Floor[n/2]}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 06 2014 *)
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PROG
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(PARI) {a(n)=n*polcoeff(sum(m=1, n+1, (1+2^m*x)^m*x^m/m)+x*O(x^n), n)}
(PARI) {a(n)=n*sum(k=0, n\2, binomial(n-k, k)*2^(k*(n-k))/(n-k))} \\ Paul D. Hanna, Apr 10 2009
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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