A318639 a(n) = Sum_{k>=0} n^k * log(k)^k / k!, rounded to nearest integer.

1, 2, 22, 646, 28847, 1741588, 133980041, 12608022914, 1409256807168, 183015824998133, 27146136664293731, 4536471294450895300, 844659618442741504695, 173611839268827045840473, 39085824299332714462271372, 9574184453657569104285899833, 2536995721294132939799176959316, 723576083578946843489853252981403, 221140244488698891750492920932788745

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 * A354243 A220732 A333796

Adjacent sequences: A318636 A318637 A318638 * A318640 A318641 A318642

Keyword

nonn

Author

Paul D. Hanna, Sep 13 2018

Status

approved