A367785
Expansion of e.g.f. exp(exp(3*x) - x - 1).
Original entry on oeis.org
1, 2, 13, 89, 772, 7745, 87949, 1109288, 15332539, 229840361, 3706130914, 63857565095, 1169261937973, 22646779177898, 462143532144937, 9902312863237637, 222119823632283628, 5202170552214520637, 126914730275907871201, 3218552632981994910248, 84686139239808135094879, 2307953474037054591248501
Offset: 0
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nmax = 21; CoefficientList[Series[Exp[Exp[3 x] - x - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] + Sum[Binomial[n - 1, k - 1] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 3^k BellB[k], {k, 0, n}], {n, 0, 21}]
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my(x='x+O('x^30)); Vec(serlaplace(exp(exp(3*x) - x - 1))) \\ Michel Marcus, Nov 30 2023
A367840
Expansion of e.g.f. 1/(2 + x - exp(4*x)).
Original entry on oeis.org
1, 3, 34, 514, 10456, 265704, 8103120, 288302480, 11722944896, 536262671488, 27256865214208, 1523936708699904, 92949383868668928, 6141694449341637632, 437033351625771001856, 33319937543640487708672, 2709708041047388536274944
Offset: 0
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-i*v[i]+sum(j=1, i, 4^j*binomial(i, j)*v[i-j+1])); v;
A367939
Expansion of e.g.f. exp(exp(4*x) - 1 - 2*x).
Original entry on oeis.org
1, 2, 20, 168, 1936, 25376, 378688, 6284928, 114471168, 2263605760, 48192279552, 1097180784640, 26562251100160, 680591327567872, 18381995707154432, 521521320660205568, 15495495061984051200, 480873815489757970432, 15549555768325162926080, 522810678067316117733376
Offset: 0
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nmax = 19; CoefficientList[Series[Exp[Exp[4 x] - 1 - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
Table[Sum[Binomial[n, k] (-2)^(n + k) BellB[k], {k, 0, n}], {n, 0, 19}]
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my(x='x+O('x^30)); Vec(serlaplace(exp(exp(4*x) - 1 - 2*x))) \\ Michel Marcus, Dec 07 2023
A367940
Expansion of e.g.f. exp(exp(4*x) - 1 - 3*x).
Original entry on oeis.org
1, 1, 17, 113, 1377, 17185, 252401, 4104721, 73500609, 1430779713, 30026750161, 674586467505, 16130795165473, 408560492670049, 10915540174130353, 306531211899158609, 9019774516570506113, 277345675943850865281, 8889954225208868308369, 296408283056785166556401
Offset: 0
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nmax = 19; CoefficientList[Series[Exp[Exp[4 x] - 1 - 3 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -3 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
Table[Sum[Binomial[n, k] (-3)^(n - k) 4^k BellB[k], {k, 0, n}], {n, 0, 19}]
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my(x='x+O('x^30)); Vec(serlaplace(exp(exp(4*x) - 1 - 3*x))) \\ Michel Marcus, Dec 07 2023
Showing 1-4 of 4 results.