A330150
Expansion of e.g.f. exp(-x) / (1 - log(1 + x)).
Original entry on oeis.org
1, 0, 0, 1, -1, 8, -16, 159, -659, 6824, -46680, 517581, -4941685, 61043344, -735256328, 10269016939, -147207286503, 2322683458544, -38298239486672, 677630804946393, -12581447014620585, 247342217288517496, -5096876494438056928, 110338442309322274295
Offset: 0
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nmax = 23; CoefficientList[Series[Exp[-x]/(1 - Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!
A291980
Triangle read by rows, T(n, k) = n!*[t^k] ([x^n] exp(x*t)/(1 - log(1+x))) for 0<=k<=n.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 4, 8, 6, 4, 1, 14, 20, 20, 10, 5, 1, 38, 84, 60, 40, 15, 6, 1, 216, 266, 294, 140, 70, 21, 7, 1, 600, 1728, 1064, 784, 280, 112, 28, 8, 1, 6240, 5400, 7776, 3192, 1764, 504, 168, 36, 9, 1
Offset: 0
Triangle starts:
[1]
[1, 1]
[1, 2, 1]
[2, 3, 3, 1]
[4, 8, 6, 4, 1]
[14, 20, 20, 10, 5, 1]
[38, 84, 60, 40, 15, 6, 1]
[216, 266, 294, 140, 70, 21, 7, 1]
[600, 1728, 1064, 784, 280, 112, 28, 8, 1]
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T_row := proc(n) exp(x*t)/(1 - log(1+x)): series(%, x, n+1):
seq(n!*coeff(coeff(%,x,n), t, k), k=0..n) end:
seq(T_row(n), n=0..10);
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T[n_, k_] := Binomial[n, n - k]*Sum[j!*StirlingS1[n - k, j], {j, 0, n - k}]; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* Detlef Meya, May 12 2024 *)
A305923
Expansion of e.g.f. exp(x)/(1 - log(1 + x) - log(1 + x)^2).
Original entry on oeis.org
1, 2, 6, 21, 105, 580, 4332, 33173, 333057, 3249334, 41175698, 485901669, 7470988137, 102962077608, 1870375878472, 29342124588357, 617978798588225, 10818920340476010, 260570216908845406, 5009431835664474101, 136578252867673635369, 2844357524328057280332, 87134882338620095240484
Offset: 0
E.g.f.: A(x) = 1 + 2*x/1! + 6*x^2/2! + 21*x^3/3! + 105*x^4/4! + 580*x^5/5! + 4332*x^6/6! + ...
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a:=series(exp(x)/(1-log(1+x)-log(1+x)^2),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
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nmax = 22; CoefficientList[Series[Exp[x]/(1 - Log[1 + x] - Log[1 + x]^2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[Binomial[n, k] StirlingS1[k, j] j! Fibonacci[j + 1], {j, 0, k}], {k, 0, n}], {n, 0, 22}]
A368288
Expansion of e.g.f. exp(2*x) / (1 - log(1+x)).
Original entry on oeis.org
1, 3, 9, 28, 92, 326, 1262, 5412, 25720, 136208, 792432, 5105376, 35369072, 271130224, 2163931408, 19516167712, 172444938240, 1853022376064, 16940180000128, 231342744007680, 1864622339520768, 39188769208491520, 160619617213475840, 9585537940543741952, -35595308731629374464
Offset: 0
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=2^i+sum(j=1, i, (-1)^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;
A368289
Expansion of e.g.f. exp(-2*x) / (1 - log(1 + x)).
Original entry on oeis.org
1, -1, 1, 0, -4, 22, -98, 508, -2952, 21040, -169360, 1579168, -16208784, 185045936, -2290934384, 30842081632, -445643595776, 6905128910976, -113892295743104, 1995421707848192, -36964967819409152, 722345322667829760, -14842592110869541888
Offset: 0
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-2)^i+sum(j=1, i, (-1)^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;
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