A158876
Expansion of e.g.f.: exp( Sum_{n>=1} (n-1)! * x^n ).
Original entry on oeis.org
1, 1, 3, 19, 217, 4041, 113611, 4532683, 244208049, 17085010897, 1504881245971, 162835665686211, 21219897528855433, 3276502399914104089, 591351260856215820507, 123322423833602768272891, 29423834155886520870184801, 7963056392690313008566254753
Offset: 0
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 217*x^4/4! +...
log(A(x)) = x + x^2 + 2!*x^3 + 3!*x^4 +...+ (n-1)!*x^n +....
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m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Laplace( Exp( &+[Factorial(j-1)*x^j: j in [1..m+2]] ) ))); // G. C. Greubel, Mar 04 2020
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m:=20; S:=series( exp(add((j-1)!*x^j, j=1..m+2)), x, m+1): seq(j!*coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 04 2020
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With[{m = 20}, CoefficientList[Series[Exp[Sum[(j-1)!*x^j, {j, m+2}]], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, Mar 04 2020 *)
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{a(n)=if(n==0,1,(n-1)!*sum(k=1,n,k!*a(n-k)/(n-k)!))}
for(n=0,20,print1(a(n),", "))
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{a(n)=n!*polcoeff(exp(sum(k=1,n,(k-1)!*x^k)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))
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m=20
def A158876_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P( exp(sum(factorial(j-1)*x^j for j in (1..m+2))) ).list()
a=A158876_list(m+1); [factorial(n)*a[n] for n in (0..m)] # G. C. Greubel, Mar 04 2020
A220754
Number of ordered triples (a,b,c) of elements of the symmetric group S_n such that the triple a,b,c generates a transitive group.
Original entry on oeis.org
1, 7, 194, 12858, 1647384, 361351560, 125116670160, 64439768489040, 47159227114392960, 47285264408385951360, 63057420721939066617600, 109118766834521171299756800, 239996135160204867851157273600, 659114500480471292127627441484800
Offset: 1
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nn=14; b=Sum[n!^3 x^n/n!,{n,0,nn}]; Drop[Range[0,nn]!CoefficientList[Series[Log[b],{x,0,nn}],x],1]
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N = 66; x = 'x + O('x^N);
egf = log(sum(n=0, N, n!^2*x^n));
gf = serlaplace(egf);
v = Vec(gf)
/* Joerg Arndt, Apr 14 2013 */
A306628
Expansion of e.g.f. log(Sum_{k>=0} (k*x)^k).
Original entry on oeis.org
1, 7, 140, 5394, 336024, 30663840, 3846629520, 634415055120, 133073632016640, 34596321663980160, 10919217470376902400, 4113179201266142515200, 1822997615295693513600000, 939158618068666181592806400, 556530141861219809204393472000
Offset: 1
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nmax = 20; Rest[CoefficientList[Series[Log[1 + Sum[(k*x)^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Mar 02 2019 *)
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N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, (k*x)^k))))
A266910
Number of size 2 subsets of S_n that generate a transitive subgroup of S_n.
Original entry on oeis.org
1, 12, 210, 5520, 206760, 10473120, 688821840, 57039171840, 5805880778880, 712594633766400, 103804864923513600, 17709509301413529600, 3498328696524626764800, 792308057159314683187200, 203965258080479292004608000, 59229266937652347633377280000, 19270409372174365076286590976000
Offset: 2
a(3) = 12 because there are 15 = binomial(3!,2) size 2 subsets of S_3 and every such subset generates a transitive subgroup of S_3 except: {(),(12)}, {(),(13)}, {(),(23)}.
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nn = 20; a = Sum[n!^2 x^n/n!, {n, 0, nn}]; Drop[(Range[0, nn]! CoefficientList[Series[Log[a], {x, 0, nn}], x] - Table[(n - 1)!, {n, 0, nn}])/2, 2]
A327547
Triangular array read by rows: T(n,k) is the number of ordered pairs of n-permutations that generate a group with exactly k orbits, 0 <= k <= n, n >= 0.
Original entry on oeis.org
1, 0, 1, 0, 3, 1, 0, 26, 9, 1, 0, 426, 131, 18, 1, 0, 11064, 2910, 395, 30, 1, 0, 413640, 92314, 11475, 925, 45, 1, 0, 20946960, 3980172, 438424, 34125, 1855, 63, 1, 0, 1377648720, 224782284, 21632436, 1550689, 84840, 3346, 84, 1, 0, 114078384000, 16158371184, 1353378284, 87036012, 4533249, 185976, 5586, 108, 1
Offset: 0
Triangle T(n,k) begins:
1;
0, 1;
0, 3, 1;
0, 26, 9, 1;
0, 426, 131, 18, 1;
0, 11064, 2910, 395, 30, 1;
0, 413640, 92314, 11475, 925, 45, 1;
T(3,2) = 9 because we have 3 ordered pairs (e,<(1,2)>), (<(1,2)>,e), (<(1,2)>,<(1,2)>) for each of the 3 transpositions in S_3.
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nn = 7; Range[0, nn]! CoefficientList[Series[Exp[u Log[Sum[n!^2 z^n/n!, {n, 0, nn}]]], {z, 0, nn}], {z, u}] // Grid
Showing 1-5 of 5 results.