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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 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.)