A354391
Expansion of e.g.f. 1/(1 + (exp(x) - 1)^2 / 2).
Original entry on oeis.org
1, 0, -1, -3, -1, 45, 269, 147, -11341, -101055, -73711, 8420247, 99423719, 87623445, -13791067291, -202300002453, -202683482821, 42194985241545, 738185254885529, 805294804942047, -216422419200618961, -4390167368672158755, -5040372451183319251
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+(exp(x)-1)^2/2)))
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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, j)*stirling(j, 2, 2)*v[i-j+1])); v;
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a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/(-2)^k);
A354393
Expansion of e.g.f. 1/(1 + (exp(x) - 1)^4 / 24).
Original entry on oeis.org
1, 0, 0, 0, -1, -10, -65, -350, -1631, -5250, 18395, 685850, 10485739, 127737610, 1336804105, 11432407350, 54280609109, -712071643930, -29671691715185, -660215774400350, -11770593620859521, -176475952496559870, -2055362595355830475, -9749893741512339250
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+(exp(x)-1)^4/24)))
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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, j)*stirling(j, 4, 2)*v[i-j+1])); v;
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a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/(-24)^k);
A354392
Expansion of e.g.f. 1/(1 + (exp(x) - 1)^3 / 6).
Original entry on oeis.org
1, 0, 0, -1, -6, -25, -70, 119, 4354, 48215, 371610, 1620839, -10665886, -388969945, -6114636710, -65181228841, -325375497726, 5950049261495, 226564100074970, 4447402833379079, 57902620204276834, 258292327155958535, -12701483290229413350
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+(exp(x)-1)^3/6)))
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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, j)*stirling(j, 3, 2)*v[i-j+1])); v;
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a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/(-6)^k);
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.