A336259
a(0) = 1; a(n) = (n!)^3 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^3.
Original entry on oeis.org
1, 1, 9, 278, 20464, 2948824, 735078968, 291153023664, 172201253334528, 145044581320046592, 167609226267379703808, 257816558769660828601344, 514890814087717253133447168, 1307445058678686737908660752384, 4146656933568759002389401276616704
Offset: 0
-
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-i)/i^3, i=1..n))
end:
a:= n-> n!^3*b(n):
seq(a(n), n=0..14); # Alois P. Heinz, Jan 04 2024
-
a[0] = 1; a[n_] := a[n] = (n!)^3 Sum[a[k]/(k! (n - k))^3, {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]
nmax = 14; CoefficientList[Series[1/(1 - PolyLog[3, x]), {x, 0, nmax}], x] Range[0, nmax]!^3
A336261
a(0) = 1; a(n) = (n!)^5 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^5.
Original entry on oeis.org
1, 1, 33, 8294, 8790208, 28436662624, 228929520628448, 3983602580423420352, 135150778123405293748224, 8262821715336263175482769408, 855516444430388524429593124012032, 142657102263368111456587968163250896896, 36753801552552818015956675623665562408714240
Offset: 0
-
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-i)/i^5, i=1..n))
end:
a:= n-> n!^5*b(n):
seq(a(n), n=0..14); # Alois P. Heinz, Jan 04 2024
-
a[0] = 1; a[n_] := a[n] = (n!)^5 Sum[a[k]/(k! (n - k))^5, {k, 0, n - 1}]; Table[a[n], {n, 0, 12}]
nmax = 12; CoefficientList[Series[1/(1 - PolyLog[5, x]), {x, 0, nmax}], x] Range[0, nmax]!^5
A336258
a(0) = 1; a(n) = (n!)^2 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^2.
Original entry on oeis.org
1, 1, 5, 58, 1208, 39476, 1861372, 119587224, 10040970816, 1067383279872, 140110136642304, 22256626639796352, 4207858001708629248, 933704296260740939520, 240293228328619963492608, 70992050129486593239246336, 23863916105454465092261412864
Offset: 0
-
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-i)/i^2, i=1..n))
end:
a:= n-> n!^2*b(n):
seq(a(n), n=0..16); # Alois P. Heinz, Jan 04 2024
-
a[0] = 1; a[n_] := a[n] = (n!)^2 Sum[a[k]/(k! (n - k))^2, {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[1/(1 - PolyLog[2, x]), {x, 0, nmax}], x] Range[0, nmax]!^2
A337677
a(0) = 1; a(n) = -(n!)^4 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^4.
Original entry on oeis.org
1, -1, 15, -1150, 277760, -164021776, 200693093392, -455136213439776, 1760342776470958080, -10907982472777142353920, 103006437933467240856354816, -1424284967682216438413265543168, 27890228890526992620507064048877568, -752281114397558490715695708227012591616
Offset: 0
-
a[0] = 1; a[n_] := a[n] = -(n!)^4 Sum[a[k]/(k! (n - k))^4, {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
nmax = 13; CoefficientList[Series[1/(1 + PolyLog[4, x]), {x, 0, nmax}], x] Range[0, nmax]!^4
-
a(n)={n!^4*polcoef(1/(1 + polylog(4,x + O(x*x^n))), n)} \\ Andrew Howroyd, Sep 15 2020
A368754
a(n) = (n!)^n * [x^n] * 1/(1 - polylog(n,x)).
Original entry on oeis.org
1, 1, 5, 278, 404768, 28436662624, 151309093659896512, 86745908552613198656020224, 7184659625769578063908866060107907072, 110866279942987479997999976181870531647691458347008, 399488258540989429698770032526869852804662313023226648081962369024
Offset: 0
Cf.
A000051,
A000142,
A007840,
A011782,
A036740,
A323339,
A323340,
A336258,
A336259,
A336260,
A336261.
-
a:= n-> n!^n*coeff(series(1/(1-polylog(n, x)), x, n+1), x, n):
seq(a(n), n=0..10);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1,
add(b(n-j, k)/j^k, j=1..n))
end:
a:= n-> n!^n*b(n$2):
seq(a(n), n=0..10);
Showing 1-5 of 5 results.