proposed

approved

editing

proposed

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

E.g.f.: exp( -LambertW(-x/(1-x)^3) )/(1-x)^2.

a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+2*k+1,n-k)/k!.

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^3))/(1-x)^2))

Cf. A367789, A377599, A377811.

allocated for Seiichi Manyama

E.g.f. satisfies A(x) = exp( x * A(x) / (1-x) ) / (1-x)^2.

1, 3, 19, 202, 3085, 61886, 1544029, 46182900, 1612759369, 64455582394, 2902794546961, 145497909334856, 8035136800888333, 484821204654219798, 31735810390729211173, 2240132583683741633116, 169624462686462529305745, 13715713402047448280358002, 1179576532854283015832748697

0,2

(PARI) a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(n+2*k+1, n-k)/k!);

allocated

nonn

Seiichi Manyama, Nov 14 2024

approved

editing

allocated for Seiichi Manyama

recycled

allocated

editing

approved

7-smooth numbers k such that k / (next 7-smooth number after k) reaches a record large value.

1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 20, 24, 27, 35, 48, 49, 63, 80, 125, 224, 2400, 4374, 250000, 78121827, 205885750000000, 2251783932057135, 363797880709171295166015625, 37252879910233655318543787489, 639471113830161648869830843378434048, 386856165263818792142417337638583138065421

1,2

Michael S. Branicky, <a href="/A377608/b377608_2.txt">Table of n, a(n) for n = 1..35</a>

(Python) # uses A002473gen() in A002473

from itertools import islice

from fractions import Fraction

def agen():

g = A002473gen(); k = next(g); record = 0

while True:

nextk = next(g); f = Fraction(k, nextk)

if f > record: yield k; record = f

k = nextk

print(list(islice(agen(), 30))) # Michael S. Branicky, Nov 03 2024

Cf. A002473, A085153.

nonn,changed

recycled

_Chengcheng Wei_, Nov 02 2024

a(24) corrected and a(26) and beyond from Michael S. Branicky, Nov 02 2024

proposed

editing