A097147
Total sum of minimum block sizes in all partitions of n-set.
Original entry on oeis.org
1, 3, 7, 21, 66, 258, 1079, 4987, 25195, 136723, 789438, 4863268, 31693715, 217331845, 1564583770, 11795630861, 92833623206, 760811482322, 6479991883525, 57256139503047, 523919025038279, 4956976879724565, 48424420955966635, 487810283307069696
Offset: 1
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g:= proc(n, i, p) option remember; `if`(n=0, (i+1)*p!,
`if`(i<1, 0, add(g(n-i*j, i-1, p+j*i)/j!/i!^j, j=0..n/i)))
end:
a:= n-> g(n$2, 0):
seq(a(n), n=1..30); # Alois P. Heinz, Mar 06 2015
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Drop[Apply[Plus,Table[nn=25;Range[0,nn]!CoefficientList[Series[Exp[Sum[ x^i/i!,{i,n,nn}]]-1,{x,0,nn}],x],{n,1,nn}]],1] (* Geoffrey Critzer, Jan 10 2013 *)
g[n_, i_, p_] := g[n, i, p] = If[n == 0, (i+1)*p!, If[i<1, 0,
Sum[g[n-i*j, i-1, p+j*i]/j!/i!^j, {j, 0, n/i}]]];
a[n_] := g[n, n, 0];
Array[a, 30] (* Jean-François Alcover, Aug 24 2021, after Alois P. Heinz *)
A097148
Total sum of maximum block sizes in all partitions of n-set.
Original entry on oeis.org
1, 3, 10, 35, 136, 577, 2682, 13435, 72310, 414761, 2524666, 16239115, 109976478, 781672543, 5814797281, 45155050875, 365223239372, 3070422740989, 26780417126048, 241927307839731, 2260138776632752, 21805163768404127, 216970086170175575, 2224040977932468379
Offset: 1
-
b:= proc(n, m) option remember; `if`(n=0, m, add(
b(n-j, max(j, m))*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n, 0):
seq(a(n), n=1..24); # Alois P. Heinz, Mar 02 2020, revised May 08 2024
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Drop[ Range[0, 22]! CoefficientList[ Series[ Sum[E^(E^x - 1) - E^Sum[x^j/j!, {j, 1, k}], {k, 0, 22}], {x, 0, 22}], x], 1] (* Robert G. Wilson v, Aug 05 2004 *)
A097146
Total sum of maximum list sizes in all sets of lists of n-set, cf. A000262.
Original entry on oeis.org
0, 1, 5, 31, 217, 1781, 16501, 172915, 1998641, 25468777, 352751941, 5292123431, 85297925065, 1472161501981, 27039872306357, 527253067633531, 10865963240550241, 236088078855319505, 5390956470528548101, 129102989125943058607, 3234053809095307670201, 84596120521251178630981, 2305894874979300173268085
Offset: 0
For n=4 we have 73 sets of lists (cf. A000262): (1234) (24 ways), (123)(4) (6*4 ways), (12)(34) (3*4 ways), (12)(3)(4) (6*2 ways), (1)(2)(3)(4) (1 way); so a(4)= 24*4+24*3+12*2+12*2+1*1 = 217.
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b:= proc(n, m) option remember; `if`(n=0, m, add(j!*
b(n-j, max(m, j))*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..25); # Alois P. Heinz, May 10 2016
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b[n_, m_] := b[n, m] = If[n == 0, m, Sum[j! b[n-j, Max[m, j]] Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := b[n, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 05 2020, after Alois P. Heinz *)
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N=50; x='x+O('x^N);
egf=exp(x/(1-x))*sum(k=1,N, (1-exp(x^k/(x-1))) );
Vec( serlaplace(egf) ) /* show terms */
Showing 1-3 of 3 results.
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