OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
a(n) = Sum_{k>=0} log(k^n)^k / k!, rounded to the nearest integer.
log(a(n)) ~ n^(n - 1/2) * log(n)^n * log(n*log(n*log(n)))^(n*log(n*log(n))) / (sqrt(2*Pi) * log(n*log(n))^(1/2 + n*log(n*log(n)))). - Vaclav Kotesovec, Sep 15 2018
EXAMPLE
The initial values of Sum_{k>=0} n^k * log(k)^k / k! begin:
n=0: 1
n=1: 1.785672099547734935860778589192856069327146674275145448...
n=2: 21.59893039935750356144319397458439503558078182702969038...
n=3: 645.8053741791930703577716806845658568790747976442247100...
n=4: 28847.12309840959482600168935775370329169251260992931745...
n=5: 1741587.903076664489270185782706726704206814310319809374...
n=6: 133980040.7674241503067515015896322884481377841596013399...
n=7: 12608022913.50110331415710392216643380159838762797570877...
n=8: 1409256807168.466379904069284286327483370824123237852285...
n=9: 183015824998133.3607705761259552467771528177897530667232...
n=10: 27146136664293731.1548378977029279237444516674554473767...
n=11: 4536471294450895299.98197621326037200309665282140191583...
n=12: 844659618442741504695.145062999869803538259503828818159...
n=13: 173611839268827045840473.323145586704343200892028060221...
n=14: 39085824299332714462271371.5306771659839726127936982072...
n=15: 9574184453657569104285899833.41979300490536053788507172...
n=16: 2536995721294132939799176959315.74691780446875099956447...
n=17: 723576083578946843489853252981403.043176513226329165540...
n=18: 221140244488698891750492920932788745.357323784096639994...
n=19: 72137405174355471782873335091418865841.8570612366704366...
n=20: 25028520511541449109504471282367224756153.9326945669108...
etc.
The logarithms of these sums begin:
n=1: 0.57979487072061663249684154367...
n=2: 3.07264379491577180724564218166...
n=3: 6.47049818002877678471502971293...
n=4: 10.2697655476713847022668879010...
n=5: 14.3703078430105664212110327292...
n=6: 18.7132013973242728184421978983...
n=7: 23.2575991874382258771020044708...
n=8: 27.9740835942448178680184355287...
n=9: 32.8405937404308231686932457755...
n=10: 37.840011135191148812742939590...
n=11: 42.958681135814003844455350022...
n=12: 48.185465401685909208633101944...
n=13: 53.511108951328944745457583734...
n=14: 58.927802083222127407206926376...
n=15: 64.428867867725650656453035594...
n=16: 70.008533383915269331668542704...
n=17: 75.661758490804776290557928228...
n=18: 81.384105159500487983924222643...
n=19: 87.171636053309378500732579385...
n=20: 93.020834621856469490292683085...
etc.
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 13 2018
STATUS
approved