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A167139 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!. 3
1, 4, 30, 292, 3497, 49488, 806504, 14860032, 305261640, 6914828176, 171186477632, 4597513706496, 133116705145408, 4133143450593536, 136981118139314688, 4826352390162440704, 180139085757269111824 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Conjecture: For all integers 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.

LINKS

Table of n, a(n) for n=0..16.

FORMULA

a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A005649(k)^2, cf. A101370. - Vladeta Jovovic, Nov 09 2009

EXAMPLE

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! + ...

PROG

(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)}

CROSSREFS

Cf. A167138, A005649.

Sequence in context: A127130 A052631 A301334 * A347994 A240958 A054972

Adjacent sequences: A167136 A167137 A167138 * A167140 A167141 A167142

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 03 2009

STATUS

approved

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Last modified February 8 01:34 EST 2023. Contains 360133 sequences. (Running on oeis4.)