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);
A354394
Expansion of e.g.f. 1/(1 + (exp(x) - 1)^5 / 120).
Original entry on oeis.org
1, 0, 0, 0, 0, -1, -15, -140, -1050, -6951, -42273, -232870, -949740, 2401399, 149618469, 2979464124, 47639256210, 683529622229, 9045426379611, 109599657976942, 1148191101672384, 8033814119097459, -50834295574038207, -3977581842278623216, -119536187842156328034
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+(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, j)*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);
A354396
Expansion of e.g.f. exp( -(exp(x) - 1)^3 / 6 ).
Original entry on oeis.org
1, 0, 0, -1, -6, -25, -80, -91, 1694, 23155, 206340, 1442969, 6928394, -6507865, -752409840, -12953182971, -160186016906, -1548849362085, -9789241693220, 28359195353489, 2378650585685794, 52832659521004495, 855581150441210600, 10878338100191146749
Offset: 0
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With[{nn=30},CoefficientList[Series[Exp[-(Exp[x]-1)^3/6],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 02 2023 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-(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-1, j-1)*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*k!));
Showing 1-4 of 4 results.