A346974
Expansion of e.g.f. log( 1 + (exp(x) - 1)^2 / 2 ).
Original entry on oeis.org
1, 3, 4, -15, -134, -357, 2374, 33645, 133186, -1288617, -24887906, -130115895, 1666879306, 40612637523, 262868197414, -4221449488635, -123802488449774, -952293015617937, 18497401668708334, 632675912865355425, 5622243546094977946, -128799294291220310997
Offset: 2
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nmax = 23; CoefficientList[Series[Log[1 + (Exp[x] - 1)^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
a[n_] := a[n] = StirlingS2[n, 2] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 2, 23}]
A346975
Expansion of e.g.f. log( 1 + (exp(x) - 1)^3 / 3! ).
Original entry on oeis.org
1, 6, 25, 80, 91, -1694, -22875, -193740, -1119569, -768394, 101162425, 1930987240, 23583202371, 181575384906, -306743537075, -45405986594980, -1070132302146089, -16439720013909794, -145808623945689375, 1048196472097011600, 84226169502099763051
Offset: 3
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nmax = 23; CoefficientList[Series[Log[1 + (Exp[x] - 1)^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
a[n_] := a[n] = StirlingS2[n, 3] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 3, 23}]
A346976
Expansion of e.g.f. log( 1 + (exp(x) - 1)^4 / 4! ).
Original entry on oeis.org
1, 10, 65, 350, 1666, 6510, 7855, -270050, -4942894, -63052990, -682650605, -6309889950, -42960995804, 348211510, 7739540496935, 202902567668150, 3863986259609686, 61527382177040010, 807717870749781475, 7066953051021894250, -33781117662453993424
Offset: 4
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nmax = 24; CoefficientList[Series[Log[1 + (Exp[x] - 1)^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
a[n_] := a[n] = StirlingS2[n, 4] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 24}]
A354398
Expansion of e.g.f. exp( -(exp(x) - 1)^5 / 120 ).
Original entry on oeis.org
1, 0, 0, 0, 0, -1, -15, -140, -1050, -6951, -42399, -239800, -1164570, -2553551, 54771717, 1384600854, 23301803070, 340911045929, 4600861076433, 58236569430172, 687816515641206, 7315220762286129, 61629305427537309, 140107851269900954, -11001310744922517426
Offset: 0
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With[{nn=30},CoefficientList[Series[Exp[-(Exp[x]-1)^5/120],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 23 2025 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-(exp(x)-1)^5/120)))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, binomial(i-1, j-1)*stirling(j, 5, 2)*v[i-j+1])); v;
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a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/((-120)^k*k!));
Showing 1-4 of 4 results.