A182928
Triangular array read by rows: [T(n,k),k=1..tau(n)] = [-n!/(d*(-(n/d)!)^d), d|n].
Original entry on oeis.org
1, 1, -1, 1, 2, 1, -3, -6, 1, 24, 1, -10, 30, -120, 1, 720, 1, -35, -630, -5040, 1, 560, 40320, 1, -126, 22680, -362880, 1, 3628800, 1, -462, 11550, -92400, -1247400, -39916800, 1, 479001600, 1, -1716, 97297200, -6227020800
Offset: 1
The array starts with
[1] 1,
[2] 1, -1,
[3] 1, 2,
[4] 1, -3, -6,
[5] 1, 24,
[6] 1, -10, 30, -120,
[7] 1, 720,
[8] 1, -35, -630, -5040,
[9] 1, 560, 40320,
-
A182928_row := proc(n) local d;
seq(-n!/(d*(-(n/d)!)^d), d = numtheory[divisors](n)) end:
-
row[n_] := Table[ -n!/(d*(-(n/d)!)^d), {d, Divisors[n]}]; Table[row[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)
A292308
E.g.f.: 1/Product_{k>=1} (1 + x^k/k!).
Original entry on oeis.org
1, -1, 1, -4, 21, -96, 520, -3795, 32053, -284368, 2763876, -30648465, 373339824, -4833294389, 67167087793, -1009753574739, 16215467043493, -275361718915824, 4947532173402532, -94054153646919213, 1882793796608183356, -39528099512321898363
Offset: 0
-
nmax = 20; Table[SeriesCoefficient[Product[1/(1 + x^k/k!), {k, 1, n}], {x, 0, n}], {n, 0, nmax}] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 14 2017 *)
A319219
Expansion of e.g.f. Product_{k>=1} 1/(1 + x^k/(k - 1)!).
Original entry on oeis.org
1, -1, 0, -3, 32, -105, 204, -3325, 52408, -376425, 1304180, -25766301, 659066484, -6675505837, 30765540974, -893416597515, 29169795361424, -380344619169729, 2379504317523300, -84225906785770525, 3388223174832010540, -55107296201168047221, 422923168260105913070
Offset: 0
-
seq(n!*coeff(series(mul(1/(1 + x^k/(k - 1)!),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019
-
nmax = 22; CoefficientList[Series[Product[1/(1 + x^k/(k - 1)!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[x^(j k)/(k (-(j - 1)!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[d (-(d - 1)!)^(-k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 22}]
A346315
Sum_{n>=0} a(n) * x^n / (n!)^2 = Product_{n>=1} 1 / (1 + (-x)^n / (n!)^2).
Original entry on oeis.org
1, 1, 3, 28, 483, 11976, 423660, 20801775, 1337182819, 108259612048, 10814058518328, 1308659192928495, 188498906179378476, 31855351764833425895, 6243218508505581436249, 1404734813476218805338303, 359618310105650201828166499, 103929494668760259335327432160
Offset: 0
-
nmax = 17; CoefficientList[Series[Product[1/(1 + (-x)^k/(k!)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[(-1)^k (Binomial[n, k] k!)^2 k Sum[(-1)^d/(d ((k/d)!)^(2 d)), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
Showing 1-4 of 4 results.
Comments