A336961
Expansion of e.g.f. exp(x * (2 + x) * exp(x)).
Original entry on oeis.org
1, 2, 10, 56, 384, 3022, 26626, 258624, 2734360, 31168682, 380196414, 4932536908, 67717987948, 979613124414, 14877703575130, 236469561581768, 3922587278751504, 67743812585483218, 1215417753459838198, 22609895367286957572, 435341977596130683316
Offset: 0
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nmax = 20; CoefficientList[Series[Exp[x (2 + x) Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] k (k + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
A347665
E.g.f.: exp( exp(x) * (1 + x + x^2 / 2) - 1 ).
Original entry on oeis.org
1, 2, 8, 39, 227, 1518, 11368, 93796, 842416, 8158942, 84581560, 932878169, 10891741957, 134043979644, 1732583270218, 23445954950207, 331260511278659, 4874617929283392, 74548457001207068, 1182551615010825076, 19423368875596930596, 329809489306236629874
Offset: 0
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nmax = 21; CoefficientList[Series[Exp[Exp[x] (1 + x + x^2/2) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (k (k + 1)/2 + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
A347666
E.g.f.: exp( exp(x) * (1 + x + x^2 / 2 + x^3 / 6) - 1 ).
Original entry on oeis.org
1, 2, 8, 40, 239, 1648, 12778, 109476, 1023520, 10341878, 112067820, 1294254184, 15847382977, 204827368606, 2784056034014, 39665514607872, 590684848605779, 9170941154737032, 148120725648168260, 2483657480026985432, 43157660169344697996, 775898068395820783674
Offset: 0
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nmax = 21; CoefficientList[Series[Exp[Exp[x] (1 + x + x^2/2 + x^3/6) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] ((k^3 + 5 k + 6)/6) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
A352376
Expansion of e.g.f. exp(1 - (1 + x) * exp(x)).
Original entry on oeis.org
1, -2, 1, 6, -2, -58, -91, 732, 4365, -1468, -140682, -685886, 1791101, 43923266, 216543097, -939472974, -22047365454, -127801626362, 541608607233, 16524264652568, 124850392700061, -279906371211584, -16968403342944782, -176737444660619046
Offset: 0
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With[{nn=30},CoefficientList[Series[Exp[1-(1+x)Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 11 2024 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(1-(1+x)*exp(x))))
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a(n) = if(n==0, 1, -sum(k=1, n, (k+1)*binomial(n-1, k-1)*a(n-k)));
A372623
Expansion of e.g.f. exp( exp(x) * (1 + x^2 / 2) - 1 ).
Original entry on oeis.org
1, 1, 3, 11, 48, 247, 1448, 9445, 67651, 526704, 4418875, 39670270, 378931567, 3832882393, 40886570975, 458341921775, 5382862509572, 66050096110691, 844741961321026, 11236481306649167, 155150031880549077, 2219877203279634396, 32860282502526114729
Offset: 0
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nmax = 22; CoefficientList[Series[Exp[Exp[x] (1 + x^2/2) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (k (k - 1)/2 + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
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