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A318639 a(n) = Sum_{k>=0} n^k * log(k)^k / k!, rounded to nearest integer. 1
1, 2, 22, 646, 28847, 1741588, 133980041, 12608022914, 1409256807168, 183015824998133, 27146136664293731, 4536471294450895300, 844659618442741504695, 173611839268827045840473, 39085824299332714462271372, 9574184453657569104285899833, 2536995721294132939799176959316, 723576083578946843489853252981403, 221140244488698891750492920932788745 (list; graph; refs; listen; history; text; internal format)
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

Sequence in context: A210657 A177042 A308535 * A220732 A333796 A132568

Adjacent sequences:  A318636 A318637 A318638 * A318640 A318641 A318642

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 13 2018

STATUS

approved

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Last modified September 23 20:42 EDT 2021. Contains 347617 sequences. (Running on oeis4.)