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Search: id:A167139
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| A167139 |
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G.f.: Sum_{n>=0} A005649(n)^2 * log(1+x)^n/n! where 1/(1-x)^2 = Sum_{n>=0} A005649(n)*log(1+x)^n/n!. |
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4
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| 1, 4, 30, 292, 3497, 49488, 806504, 14860032, 305261640, 6914828176, 171186477632, 4597513706496, 133116705145408, 4133143450593536, 136981118139314688, 4826352390162440704, 180139085757269111824
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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CONJECTURE: For all integer m>0, Sum_{n>=0} L(n)^m * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.
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FORMULA
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a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A005649(k)^2, cf. A101370. [From Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 09 2009]
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EXAMPLE
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G.f.: A(x) = 1 + 4*x + 30*x^2 + 292*x^3 + 3497*x^4 + 49488*x^5 +...
Illustrate A(x) = Sum_{n>=0} A005649(n)^2 * log(1+x)^n/n!:
A(x) = 1 + 2^2*log(1+x) + 8^2*log(1+x)^2/2! + 44^2*log(1+x)^3/3! + 308^2*log(1+x)^4/4! + 2612^2*log(1+x)^5/5! +...+ A005649(n)^2*log(1+x)^n/n! +...
where the g.f. of A005649 is 1/(2 - exp(x))^2:
1/(1-x)^2 = 1 + 2*log(1+x) + 8*log(1+x)^2/2! + 44*log(1+x)^3/3! + 308*log(1+x)^4/4! + 2612*log(1+x)^5/5! +...+ A005649(n)*log(1+x)^n/n! +...
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PROGRAM
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(PARI) {Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{A005649(n)=sum(k=0, n, (k+1)*Stirling2(n, k)*k!)}
{a(n)=polcoeff(sum(m=0, n, A005649(m)^2*log(1+x+x*O(x^n))^m/m!), n)}
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CROSSREFS
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Cf. A167138, A005649.
Sequence in context: A052658 A127130 A052631 this_sequence A054972 A052452 A088794
Adjacent sequences: A167136 A167137 A167138 this_sequence A167140 A167141 A167142
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Nov 03 2009
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