A183230
G.f.: Sum_{n>=0} a(n)*x^n/n!^3 = Product_{n>=1} (1 + x^n/n!^3).
Original entry on oeis.org
1, 1, 1, 28, 65, 1126, 219592, 1210105, 26891713, 2147043538, 2019029825126, 21746314187335, 770200602942872, 54021095931416459, 16833586753169817373, 54446959965626243089903, 1039787297277083116535233
Offset: 0
G.f.: A(x) = 1 + x + x^2/2!^3 + 28*x^3/3!^3 + 65*x^4/4!^3 +...
A(x) = (1 + x)*(1 + x^2/2!^3)*(1 + x^3/3!^3)*(1 + x^4/4!^3)*...
A336294
a(n) = (n!)^n * [x^n] Product_{k>=1} (1 + x^k/(k!)^n).
Original entry on oeis.org
1, 1, 1, 28, 257, 103126, 46667437282, 140776183474585, 38414859209967468545, 8006615289848673023223926602, 100856872226698664486645150126408916015626, 7425498079138047573566961707334890995112470771975
Offset: 0
-
Table[(n!)^n SeriesCoefficient[Product[(1 + x^k/(k!)^n), {k, 1, n}], {x, 0, n}], {n, 0, 11}]
A346314
Sum_{n>=0} a(n) * x^n / (n!)^2 = Product_{n>=1} (1 - x^n / (n!)^2).
Original entry on oeis.org
1, -1, -1, 8, 15, 124, -3340, -9311, -102641, -1880812, 150047424, 692058289, 8916106452, 167039809897, 7435628931289, -1381243302601067, -9407162843960561, -165954439670564988, -3103870029424074136, -123659189880256295879, -10671656695397289496160
Offset: 0
-
nmax = 20; CoefficientList[Series[Product[(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[(Binomial[n, k] k!)^2 k Sum[1/(d ((k/d)!)^(2 d)), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
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.