A076900
Expansion of e.g.f.: 1/Product_{m>0} (1-x^m/(m-1)!).
Original entry on oeis.org
1, 1, 4, 15, 88, 505, 4056, 31549, 311816, 3083049, 36343720, 431215741, 5937234348, 82236865165, 1291252453050, 20477737537755, 361495828272496, 6449450737736065, 126566562342343176, 2509520619696338269, 54179963857121953460, 1182248224137860933781
Offset: 0
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(n, i)*i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, May 11 2016
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b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i-1] + If[i > n, 0, b[n-i, i] Binomial[n, i] i]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 03 2020, after Alois P. Heinz *)
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)!)))))}
A319218
Expansion of e.g.f. Product_{k>=1} (1 - x^k/(k - 1)!).
Original entry on oeis.org
1, -1, -2, 3, 8, 75, -216, -175, -3816, -36225, 189800, 325149, 2375460, 25547951, 386162126, -3290670825, -6316583056, -59290501809, -310987223208, -4836373835707, -86500419684420, 1119358992256239, 3043733432729198, 26408738842522959, 169835931388147464
Offset: 0
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seq(n!*coeff(series(mul((1 - x^k/(k - 1)!),k=1..100),x=0,25),x,n),n=0..24); # Paolo P. Lava, Jan 09 2019
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nmax = 24; CoefficientList[Series[Product[(1 - x^k/(k - 1)!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Exp[-Sum[Sum[x^(j k)/(k (j - 1)!^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[-d (d - 1)!^(-k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 24}]
A319219
Expansion of e.g.f. Product_{k>=1} 1/(1 + x^k/(k - 1)!).
Original entry on oeis.org
1, -1, 0, -3, 32, -105, 204, -3325, 52408, -376425, 1304180, -25766301, 659066484, -6675505837, 30765540974, -893416597515, 29169795361424, -380344619169729, 2379504317523300, -84225906785770525, 3388223174832010540, -55107296201168047221, 422923168260105913070
Offset: 0
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seq(n!*coeff(series(mul(1/(1 + x^k/(k - 1)!),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019
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nmax = 22; CoefficientList[Series[Product[1/(1 + x^k/(k - 1)!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[x^(j k)/(k (-(j - 1)!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[d (-(d - 1)!)^(-k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 22}]
A371310
Expansion of e.g.f. Product_{k>=1} (1 + prime(k)*x^k/k!).
Original entry on oeis.org
1, 2, 3, 23, 47, 231, 2260, 6527, 35151, 224759, 3434124, 12476055, 79758206, 491191521, 4752819625, 105146082344, 393097093065, 2976053272527, 21569670506914, 188844207315245, 2277243901499454, 72603521472295945, 326137558352646889, 2491611720654851668
Offset: 0
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nmax = 23; CoefficientList[Series[Product[(1 + Prime[k] x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
A371311
Expansion of e.g.f. Product_{k>=1} (1 + k*x^k/(k-1)!).
Original entry on oeis.org
1, 1, 4, 21, 52, 465, 3306, 14161, 74208, 960777, 10558630, 44851521, 361716576, 2473446157, 46951741760, 735722365995, 3502764883456, 27660533205537, 257573937401838, 2415069153393553, 62591287234200960, 1356650271603527061, 6966660193683272104, 61046400429116180475
Offset: 0
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nmax = 23; CoefficientList[Series[Product[(1 + k x^k/(k - 1)!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
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