A221176
a(n) = Sum_{i=0..n} Stirling2(n,i)*2^(4i).
Original entry on oeis.org
1, 16, 272, 4880, 91920, 1810192, 37142288, 791744272, 17490370320, 399558315792, 9421351690000, 228916588400400, 5723078052339472, 147025755978698512, 3876566243300318992, 104789417805394595600, 2901159958960121863952, 82188946843192555474704, 2380551266738846355103504, 70441182699006212824911632
Offset: 0
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With[{nn=20},CoefficientList[Series[Exp[16 (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 19 2024 *)
A345078
a(0) = 1; a(n) = 7 * Sum_{k=1..n} binomial(n,k) * a(k-1).
Original entry on oeis.org
1, 7, 63, 609, 6349, 70693, 835051, 10408335, 136290371, 1867933865, 26712000161, 397487932457, 6140285212915, 98264596199651, 1626101133819855, 27779382241071769, 489188555650420493, 8867962363328434205, 165284825277198034611, 3163858565498874214559, 62133992974174011252635
Offset: 0
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a[0] = 1; a[n_] := a[n] = 7 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1 + 7 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
A299824
a(n) = (1/e^n)*Sum_{j >= 1} j^n * n^j / (j-1)!.
Original entry on oeis.org
2, 22, 309, 5428, 115155, 2869242, 82187658, 2661876168, 96202473183, 3838516103310, 167606767714397, 7949901069639228, 407048805012563038, 22376916254447538882, 1314573505901491675965, 82188946843192555474704, 5448870914168179374456623, 381819805747937892412056342
Offset: 1
a(4) = (1/e^4)*Sum_{j >= 1} j^4 * 4^j / (j-1)! = 5428.
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a(n) = round(exp(-n)*suminf(j = 1, (j^n)*(n^j)/(j-1)!)); \\ Michel Marcus, Feb 24 2018
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A299824(n,f=exp(n),S=n/f,t)=for(j=2,oo,S+=(t=j^n*n^j)/(f*=j-1);tn&&return(ceil(S))) \\ For n > 23, use \p## with some ## >= 2n. - M. F. Hasler, Mar 09 2018
A276506
E.g.f.: exp(9*(exp(x)-1)).
Original entry on oeis.org
1, 9, 90, 981, 11511, 144108, 1911771, 26730981, 392209380, 6016681467, 96202473183, 1599000785730, 27563715220509, 491777630207037, 9064781481234546, 172346601006842337, 3375007346801025099, 67983454804021156548, 1406921223577401454239, 29881379179971835132761
Offset: 0
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a:= proc(n) option remember; `if`(n=0, 1,
(1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*9)
end:
seq(a(n), n=0..25); # Alois P. Heinz, Sep 25 2017
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Table[BellB[n, 9], {n, 0, 30}]
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my(x='x+O('x^99)); Vec(serlaplace(exp(9*(exp(x)-1)))) \\ Altug Alkan, Sep 17 2016
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