A379933
Expansion of e.g.f. 1/( exp(-x) - x )^2.
Original entry on oeis.org
1, 4, 22, 158, 1408, 15002, 186100, 2634998, 41937136, 741170834, 14402727484, 305225470046, 7005711916840, 173134991854970, 4583675648417044, 129424786945875398, 3882446011526729440, 123304773913531035170, 4133369745467043807340, 145840627118145774415214
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x)-x)^2))
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a(n) = n!*sum(k=0, n, (k+1)*(k+2)^(n-k)/(n-k)!);
A379995
Expansion of e.g.f. exp(-2*x)/(exp(-x) - x)^4.
Original entry on oeis.org
1, 6, 48, 476, 5608, 76372, 1179016, 20332580, 387225120, 8068825988, 182564048824, 4456476380644, 116724944900272, 3264981100202564, 97130013288324552, 3062011655207131748, 101963095705628194624, 3576126056313566090500, 131762871920106615643480
Offset: 0
A358740
Expansion of Sum_{k>=0} k! * ( k * x/(1 - k*x) )^k.
Original entry on oeis.org
1, 1, 9, 195, 7699, 482309, 43994741, 5508667927, 906931827831, 189998213001033, 49359340639141993, 15573690455085072011, 5866304418414451865723, 2600416934781350100016717, 1340037064604153376788884701, 794358527033920600533985973631
Offset: 0
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nmax = 20; CoefficientList[1 + Series[Sum[k! * (k * x/(1 - k*x))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 18 2023 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*(k*x/(1-k*x))^k))
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a(n) = if(n==0, 1, sum(k=1, n, k!*k^n*binomial(n-1, k-1)));
A358741
Expansion of Sum_{k>=0} k! * ( k * x/(1 - x) )^k.
Original entry on oeis.org
1, 1, 9, 179, 6655, 400581, 35530421, 4357960999, 706230728379, 146116931998025, 37577989723572001, 11758017370126904091, 4398121660346674034039, 1938019214715102033590029, 993580299268226843514372045, 586357970017371399763899232271
Offset: 0
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nmax = 20; CoefficientList[1 + Series[Sum[k! * (k * x/(1 - x))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 18 2023 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*(k*x/(1-x))^k))
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a(n) = if(n==0, 1, sum(k=1, n, k!*k^k*binomial(n-1, k-1)));
A379992
Expansion of e.g.f. exp(-3*x)/(exp(-x) - x)^2.
Original entry on oeis.org
1, 1, 7, 41, 349, 3539, 42451, 585605, 9130297, 158692679, 3041499871, 63712004729, 1447946191957, 35479218963083, 932326476195115, 26153289728300909, 779995883104560241, 24644267406802467215, 822278654588440803511, 28891372907012629446881
Offset: 0
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With[{nn=20},CoefficientList[Series[Exp[-3x]/(Exp[-x]-x)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 14 2025 *)
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a(n) = n!*sum(k=0, n, (k+1)*(k-1)^(n-k)/(n-k)!);
A379997
Expansion of e.g.f. 1/(exp(x) - x*exp(2*x))^2.
Original entry on oeis.org
1, 0, 6, 22, 224, 2138, 25732, 351846, 5458224, 94441042, 1803255404, 37652268014, 853321021192, 20858236815258, 546941712302052, 15313467390967222, 455933682027961184, 14383416438784605602, 479254037890010238172, 16817855455956128823486, 619953003446894086537656
Offset: 0
Showing 1-6 of 6 results.