A386254
Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times.
Original entry on oeis.org
1, 1, 2, 6, 18, 60, 240, 1085, 5012, 23730, 121440, 685707, 4144668, 25614589, 159141892, 1012740885, 6805631232, 48872707006, 369227821608, 2853779791619, 22131042288980, 172055270717463, 1362017827326860, 11208504802237327, 96939147303239304, 875473007351905045
Offset: 0
a(3) = 6 counts: (1,1,1), (1,2,2), (2,1,2), (2,2,1), (2,2,2), (3,3,3).
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+add(b(n-j, min(n-j, i-1))/j!, j=i..n)))
end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..25); # Alois P. Heinz, Jul 17 2025
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terms=26; CoefficientList[Series[Product[1+Sum[x^j/j!, {j,k,terms}],{k,terms}],{x,0,terms-1}],x]Range[0,terms-1]! (* Stefano Spezia, Jul 17 2025 *)
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D_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + sum(i=k,N, x^i/i!))))}
A386255
Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times and exactly one of each kind of letter is marked.
Original entry on oeis.org
1, 1, 4, 15, 64, 325, 1776, 11179, 72640, 489969, 3435580, 26495491, 221599104, 1893705697, 16145571820, 138299146665, 1241234863936, 12033569772769, 124055067568788, 1303750295285563, 13577876900409280, 139418829477000801, 1441311794301705964, 15537427948684769425
Offset: 0
a(3) = 15 counts: (1#,1,1), (1,1#,1), (1,1,1#), (1#,2#,2), (1#,2,2#), (2#,1#,2), (2,1#,2#), (2#,2,1#), (2,2#,1#), (2#,2,2), (2,2#,2), (2,2,2#), (3#,3,3), (3,3#,3), (3,3,3#) where # denotes a mark.
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+add(b(n-j, min(n-j, i-1))/(j-1)!, j=i..n)))
end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..23); # Alois P. Heinz, Jul 17 2025
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terms=24; CoefficientList[Series[Product[1+Sum[x^j/(j-1)!, {j,k,terms}],{k,terms}],{x,0,terms-1}],x]Range[0,terms-1]! (* Stefano Spezia, Jul 17 2025 *)
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E_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + sum(i=k,N, x^i/((i-1)!)))))}
A347005
E.g.f.: Product_{k>=1} 1 / (1 - exp(x) * x^k / k!).
Original entry on oeis.org
1, 1, 5, 28, 205, 1856, 19964, 249005, 3535613, 56339884, 996009280, 19350090365, 409850078356, 9400728524669, 232154433941057, 6141705628777193, 173295665869432733, 5195039603196754564, 164890990869273983108, 5524278740902526776085, 194815729875439415542760
Offset: 0
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nmax = 20; CoefficientList[Series[Product[1/(1 - Exp[x] x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
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