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);
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);
A354395
Expansion of e.g.f. exp( -(exp(x) - 1)^2 / 2 ).
Original entry on oeis.org
1, 0, -1, -3, -4, 15, 149, 672, 1091, -12855, -162796, -1060653, -2925319, 30881760, 598929239, 5688937797, 29126981516, -112222099065, -4930674413971, -69798552313728, -598032658869829, -1296500625378255, 65193402297999524, 1515140106814565547
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-(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-1, j-1)*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*k!));
A354389
Expansion of e.g.f. 1/(1 + log(1 + x)^2 / 2).
Original entry on oeis.org
1, 0, -1, 3, -5, -10, 146, -756, 1086, 23400, -300066, 1855590, 341826, -165915828, 2158958556, -10006622640, -172337345496, 4941605486016, -64365944851512, 339328464492456, 5510899593823176, -157099566384759600, 1059259019507498160, 41957473280879898720
Offset: 0
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With[{nn=30},CoefficientList[Series[1/(1+(Log[1+x]^2)/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 08 2024 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1+x)^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, 1)*v[i-j+1])); v;
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a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 1)/(-2)^k);
Showing 1-5 of 5 results.