A352071
Expansion of e.g.f. 1 / (1 + log(1 - 4*x) / 4).
Original entry on oeis.org
1, 1, 6, 62, 904, 16984, 390128, 10586736, 331267200, 11738697600, 464539452672, 20302660659456, 971106358760448, 50452643588275200, 2829000818124208128, 170271405502300207104, 10948525752699316371456, 748994717201835804033024, 54315931193865932254543872
Offset: 0
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nmax = 18; CoefficientList[Series[1/(1 + Log[1 - 4 x]/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! (-4)^(n - k), {k, 0, n}], {n, 0, 18}]
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my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-4*x)/4))) \\ Michel Marcus, Mar 02 2022
A352074
a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-n)^(n-k).
Original entry on oeis.org
1, 1, 4, 42, 904, 34070, 2019888, 174588120, 20804747136, 3276218158560, 659664288364800, 165425062846302336, 50574549124825998336, 18520126461205806360144, 8003819275469728355033088, 4031020344281171589447408000, 2340375822778055527109749211136
Offset: 0
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Unprotect[Power]; 0^0 = 1; Table[Sum[StirlingS1[n, k] k! (-n)^(n - k), {k, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! SeriesCoefficient[1/(1 + Log[1 - n x]/n), {x, 0, n}], {n, 1, 16}]]
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a(n) = sum(k=0, n, stirling(n, k, 1)*k!*(-n)^(n-k)); \\ Michel Marcus, Mar 02 2022
A354750
Expansion of e.g.f. 1 / (1 - log(1 + 3*x) / 3).
Original entry on oeis.org
1, 1, -1, 6, -48, 534, -7542, 129240, -2603736, 60292512, -1577546928, 46021512096, -1480976147664, 52110720451152, -1990258155061776, 81995762243700864, -3624527727510038784, 171109526616468957312, -8591991935936929932672, 457246520477143117555968
Offset: 0
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nmax = 19; CoefficientList[Series[1/(1 - Log[1 + 3 x]/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! 3^(n - k), {k, 0, n}], {n, 0, 19}]
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my(x='x + O('x^20)); Vec(serlaplace(1/(1-log(1+3*x)/3))) \\ Michel Marcus, Jun 06 2022
A367852
Expansion of e.g.f. 1/(1 - x + log(1 - 3*x)/3).
Original entry on oeis.org
1, 2, 11, 102, 1320, 21804, 436986, 10283580, 277697304, 8458929792, 286825214592, 10712216384352, 436859348261904, 19313926491051360, 920053448561989296, 46977842202096405024, 2559387620091962391552, 148187802162935002975488
Offset: 0
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 3^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;
A377803
Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 - log(1-3*x) / 3) ).
Original entry on oeis.org
1, 1, 5, 51, 798, 16914, 453294, 14704542, 560274336, 24529661568, 1213557885720, 66956662956600, 4076440417367856, 271472012197547472, 19631093304600307152, 1531919987372848152240, 128314172533501646058240, 11482569303348317402868480, 1093343670892117401737893632
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1-log(1-3*x)/3))/x))
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a(n) = n!*sum(k=0, n, 3^(n-k)*abs(stirling(n, k, 1))/(n-k+1)!);
Showing 1-5 of 5 results.