A347916
E.g.f.: Product_{k>=1} (1 + x^k)^exp(-x).
Original entry on oeis.org
1, 1, 0, 6, 6, 75, 1025, 1225, 43988, 471345, 5084387, 40870181, 866782774, 8473297261, 165871287465, 3934845305287, 23390789927784, 956832091069057, 21869141108144439, 269518811758178785, 8437830353620298346, 220696789738463945981, 3231280243441039496181, 125072102239522472394691
Offset: 0
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N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+x^k)^exp(-x))))
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N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(-x)*sum(k=1, N, sigma(k>>valuation(k, 2))*x^k/k))))
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N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(-x)*sum(k=1, N, x^k/(k*(1-x^(2*k)))))))
A386497
Number of sets of lists of [n] such that one list is the largest.
Original entry on oeis.org
1, 1, 2, 12, 60, 440, 3390, 33852, 338072, 4116240, 51776730, 736751180, 11075784852, 183142075272, 3157190863190, 59336602681020, 1164223828582320, 24348331444705952, 533422896546272562, 12365952739192923660, 298208300418298756460, 7570420981014167756760
Offset: 0
a(3) = 12 counts: {(1),(2,3)}, {(1),(3,2)}, {(1,2),(3)}, {(1,3),(2)}, {(2),(3,1)}, {(2,1),(3)}, {(1,2,3)}, {(1,3,2)}, {(2,1,3)}, {(2,3,1)}, {(3,1,2)}, {(3,2,1)}.
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b:= proc(n, m, t) option remember; `if`(n=0, t, add(b(n-j, max(m, j),
`if`(j>m, 1, `if`(j=m, 0, t)))*(n-1)!*j/(n-j)!, j=1..n))
end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..21); # Alois P. Heinz, Jul 23 2025
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With[{m = 21}, CoefficientList[Series[1 + Sum[x^j*Exp[(x - x^j)/(1 - x)], {j, 1, m}], {x, 0, m}], x] * Range[0, m]!] (* Amiram Eldar, Jul 24 2025 *)
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B_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(1+sum(j=1,N, x^j*exp((x-x^j)/(1-x)))))}
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