A281997
a(n) = (n-1)^n * n^n.
Original entry on oeis.org
0, 4, 216, 20736, 3200000, 729000000, 230539333248, 96717311574016, 51998697814228992, 34867844010000000000, 28531167061100000000000, 27982542656501458535448576, 32405578483833249047003529216, 43752153272681450786889799450624
Offset: 1
A304870
L.g.f.: log(Product_{k>=1} (1 + x^k)^(k^k)) = Sum_{n>=1} a(n)*x^n/n.
Original entry on oeis.org
1, 7, 82, 1015, 15626, 279862, 5764802, 134216695, 3486784483, 99999984382, 3138428376722, 106993205100070, 3937376385699290, 155568095552047430, 6568408355712906332, 295147905179218607095, 14063084452067724991010, 708235345355334189853093, 37589973457545958193355602
Offset: 1
L.g.f.: L(x) = x + 7*x^2/2 + 82*x^3/3 + 1015*x^4/4 + 15626*x^5/5 + 279862*x^6/6 + 5764802*x^7/7 + 134216695*x^8/8 + 3486784483*x^9/9 + ...
exp(L(x)) = 1 + x + 4*x^2 + 31*x^3 + 289*x^4 + 3495*x^5 + 51268*x^6 + 891152*x^7 + 17926913*x^8 + 409907600*x^9 + ... + A261053(n)*x^n + ...
-
nmax = 19; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^k^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 19; Rest[CoefficientList[Series[Sum[k^(k + 1) x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Sum[(-1)^(n/d + 1) d^(d + 1), {d, Divisors[n]}]; Table[a[n], {n, 1, 19}]
A306377
a(n) = n^(n+1) - Sum_{k=1..n-1} k^(k+1).
Original entry on oeis.org
1, 7, 72, 934, 14511, 263197, 5468126, 128156252, 3346505197, 96372936395, 3034801313116, 103751149938746, 3827141124879891, 151520483911293537, 6408792648508559714, 288419881116435604320, 13761208522825454943617, 693870384974027681319231
Offset: 1
a(5) = 5^6 - 4^5 - 3^4 - 2^3 - 1^2 = 14511.
A341330
a(n) = Sum_{k=1..n} (-k)^(k+1).
Original entry on oeis.org
1, -7, 74, -950, 14675, -265261, 5499540, -128718188, 3358066213, -96641933787, 3041786442934, -103951418936138, 3833424966763151, -151734670591049073, 6416673685121841552, -288731231494230984304, 13774353220573494006705, -694460992134764182350927
Offset: 1
-
Accumulate[(-#)^(#+1)&/@Range[17]]
-
a(n) = sum(i=1, n, (-i)^(i+1));
-
sum = 0
for i in range(1,20):
sum += (-i)**(i+1)
print(sum, end = ", ")
A344744
a(n) is the n-th power of the concatenation of the integers from 0 through n-1.
Original entry on oeis.org
0, 1, 1728, 228886641, 2861381721051424, 3539537889086624823140625, 437104634676747795452235896466702336, 5396563761318393964062660689603780554533710504641, 6662458388479360230805308787387369820914640828074410829911019008
Offset: 1
a(1) = 0^1 = 0;
a(2) = 01^2 = 1;
a(3) = 012^3 = 1728;
a(4) = 0123^4 = 228886641.
-
a[n_] := FromDigits[Join @@ IntegerDigits @ Range[0, n - 1]]^n; Array[a, 9] (* Amiram Eldar, May 29 2021 *)
-
def a(n): return int("".join(str(i) for i in range(n)))**n
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, May 29 2021
A345360
a(n) = n^n*n - n.
Original entry on oeis.org
0, 0, 6, 78, 1020, 15620, 279930, 5764794, 134217720, 3486784392, 99999999990, 3138428376710, 106993205379060, 3937376385699276, 155568095557812210, 6568408355712890610, 295147905179352825840, 14063084452067724990992, 708235345355337676357614, 37589973457545958193355582
Offset: 0
A373004
a(n) = Sum_{k=1..n} sigma( (n/gcd(k,n))^n ).
Original entry on oeis.org
1, 8, 81, 1054, 15625, 279936, 5764801, 134480378, 3486843447, 100000000000, 3138428376721, 107006262884442, 3937376385699289, 155568095557812224, 6568408355712890625, 295150156996346511346, 14063084452067724991009, 708235345964697414795264
Offset: 1