A079600
a(n) = A000670(p-1)/p with p = prime(n+1).
Original entry on oeis.org
1, 15, 669, 9295233, 2160889815, 312685569528315, 178186034908255017, 111949757382747408023661, 217157312584485035638564618459815, 367857057871350983346531103102738773, 3897277863558255935901648057010997772527380815
Offset: 1
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N:= 60: # to use primes <= N
M:= numtheory:-pi(N):
L:= [seq(ithprime(i+1)-1, i=1..M-1)]:
S:= series(1/(2-exp(x)), x=0, N+1):
seq(coeff(S,x,L[i])*L[i]!/(L[i]+1), i=1..M-1); # Robert Israel, Mar 30 2016
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Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Fubini[0, 1] = 1; a[p_] := Fubini[p-1, 1]/p; Table[ a[p], {p, Prime[Range[2, 11]]}] (* Jean-François Alcover, Mar 30 2016 *)
A087301
a(n) = n!*Sum_{i=1..n-1} (-1)^(i+1)/i.
Original entry on oeis.org
2, 3, 20, 70, 564, 3108, 30624, 230256, 2705760, 25771680, 352805760, 4067556480, 63651813120, 861371884800, 15176802816000, 235775183616000, 4620563523072000, 81032645804544000, 1748700390205440000
Offset: 2
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Rest[Table[n!Sum[(-1)^(i+1)/i,{i,n-1}],{n,20}]] (* Harvey P. Dale, Oct 24 2011 *)
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a(n)=if(n<0,0,n!*polcoeff(log(1+x+x*O(x^n))*x/(1-x),n))
Original entry on oeis.org
0, 1, 3, 25, 340, 7026, 204862, 8007602, 404077632, 25569505628, 1982619985192, 184861494417920, 20406183592852460, 2631875641089358912, 392163247878318070876, 66855512799464487146588, 12929525365915201064027856, 2815456378791384288128303192
Offset: 1
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{ (matrix(30,30,i,j,(-1)^(i!=j)*stirling(i,j,2))^(-1))[,2] } \\ Max Alekseyev, Jun 17 2011
A293860
a(0) = 0, a(1) = 1; a(n) = n! * [x^n] exp(n*x)*Sum_{k=1..n-1} a(k)*x^k/k!.
Original entry on oeis.org
0, 1, 4, 63, 1648, 65075, 3629196, 272106555, 26418426560, 3225539263995, 483800514119500, 87459323696213843, 18755503692216214320, 4707783117485450859987, 1367396879443428912151724, 455052324991418691450493275, 172012620929344322616321833728
Offset: 0
E.g.f. A(x) = x + 4*x^2/2! + 63*x^3/3! + 1648*x^4/4! + 65075*x^5/5! + 3629196*x^6/6! + ...
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a[n_] := a[n] = n! SeriesCoefficient[Exp[n x] Sum[a[k] x^k/k!, {k, 1, n - 1}], {x, 0, n}]; a[0] = 0; a[1] = 1; Table[a[n], {n, 0, 16}]
A308475
a(1) = 1; a(n) = Sum_{k=1..n-1, gcd(n,k) = 1} binomial(n,k)*a(k).
Original entry on oeis.org
1, 2, 9, 40, 315, 1896, 21651, 191360, 2546487, 28064080, 488517183, 5879603280, 124673371719, 1928346159572, 42684093159480, 754925802649360, 20289814995554811, 366300418631427144, 11352374441063693655, 250187625076714423520, 7774760839170720287739
Offset: 1
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a:= proc(n) option remember;
if n=1 then 1;
else add( `if`(gcd(n,j)=1, binomial(n,j)*a(j), 0), j=1..n-1);
end if; end proc;
seq(a(n), n = 1..30); # G. C. Greubel, Mar 08 2021
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a[n_] := Sum[If[GCD[n, k] == 1, Binomial[n, k] a[k], 0], {k, 1, n - 1}]; a[1] = 1; Table[a[n], {n, 1, 21}]
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@CachedFunction
def a(n):
if n==1: return 1
else: return sum( kronecker_delta(gcd(n,j), 1)*binomial(n,j)*a(j) for j in (1..n-1) )
[a(n) for n in (1..30)] # G. C. Greubel, Mar 08 2021
A111154
a(n) = sum(k=0,n,B_k*A000629(k)*A000629(n-k)) where B_k is the k-th Bernoulli number.
Original entry on oeis.org
1, 1, 5, 22, 125, 948, 8627, 86618, 970649, 13105048, 206672615, 3084750894, 39005776445, 972651041372, 45030832321499, 489627529809250, -65060172242461567, 100399556385989760, 378285729309697789679, 921583343151423434486, -2826260367937145199562267
Offset: 0
A352863
a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n+3,k+3) * a(k).
Original entry on oeis.org
1, 4, 30, 260, 2625, 30296, 393372, 5675160, 90062775, 1559197420, 29242803018, 590638256572, 12781663255725, 295040675093360, 7236113219901240, 187911083837928048, 5150869386839932995, 148622674413214927140, 4502761102131604279590, 142914444471765753144820
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 3, k + 3] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[D[x^3/(3! (2 - Exp[x])), {x, 3}], {x, 0, nmax}], x] Range[0, nmax]!
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