A301655
a(n) = [x^n] 1/(1 - Sum_{k>=1} k^n*x^k).
Original entry on oeis.org
1, 1, 5, 44, 723, 24655, 1715816, 239697569, 69557364821, 41297123651644, 49900451628509015, 125141540794392423599, 641579398300246011553552, 6729809577032172543373047313, 146355880526667013027682326650073, 6505380999057202235872595196799580684
Offset: 0
-
Table[SeriesCoefficient[1/(1 - Sum[k^n x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 - PolyLog[-n, x]), {x, 0, n}], {n, 0, 15}]
A369016
Triangle read by rows: T(n, k) = binomial(n - 1, k - 1) * (k - 1)^(k - 1) * (n - k) * (n - k + 1)^(n - k - 1).
Original entry on oeis.org
0, 0, 0, 0, 1, 0, 0, 6, 2, 0, 0, 48, 18, 12, 0, 0, 500, 192, 144, 108, 0, 0, 6480, 2500, 1920, 1620, 1280, 0, 0, 100842, 38880, 30000, 25920, 23040, 18750, 0, 0, 1835008, 705894, 544320, 472500, 430080, 393750, 326592, 0
Offset: 0
Triangle starts:
[0] [0]
[1] [0, 0]
[2] [0, 1, 0]
[3] [0, 6, 2, 0]
[4] [0, 48, 18, 12, 0]
[5] [0, 500, 192, 144, 108, 0]
[6] [0, 6480, 2500, 1920, 1620, 1280, 0]
[7] [0, 100842, 38880, 30000, 25920, 23040, 18750, 0]
[8] [0, 1835008, 705894, 544320, 472500, 430080, 393750, 326592, 0]
A368849,
A368982 and this sequence are alternative sum representation for
A001864 with different normalizations.
T(n, k) =
A368849(n, k) / n for n >= 1.
T(n, n - 1) =
A055897(n - 1) for n >= 2.
Sum_{k=0..n} T(n, k) =
A000435(n) for n >= 1.
Sum_{k=0..n} (-1)^(k+1)*T(n, k) =
A368981(n) / n for n >= 1.
-
T := (n, k) -> binomial(n-1, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k-1):
seq(seq(T(n, k), k = 0..n), n=0..9);
-
A369016[n_, k_] := Binomial[n-1, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k-1); Table[A369016[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 28 2024 *)
-
def T(n, k): return binomial(n-1, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k-1)
for n in range(0, 9): print([T(n, k) for k in range(n + 1)])
A203423
a(n) = w(n+1)/(2*w(n)), where w=A203422.
Original entry on oeis.org
-3, 24, -250, 3240, -50421, 917504, -19131876, 450000000, -11789738455, 340545503232, -10752962364222, 368510430439424, -13623365478515625, 540431955284459520, -22899384412078526344, 1032236014321051140096, -49323481720063219673451, 2490368000000000000000000, -132484966403310261255807810
Offset: 1
-
[(-1)^n*(n+1)*(n+2)^n/2: n in [1..20]]; // G. C. Greubel, Dec 07 2023
-
(* First program *)
f[j_] := 1/(j + 1); z = 16;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
1/Table[v[n], {n, 1, z}] (* A203422 *)
Table[v[n]/(2 v[n + 1]), {n, 1, z}] (* this sequence *)
(* Second program *)
Table[(-1)^n*(n+1)*(n+2)^n/2, {n,20}] (* G. C. Greubel, Dec 07 2023 *)
-
[(-1)^n*(n+1)*(n+2)^n/2 for n in range(1,21)] # G. C. Greubel, Dec 07 2023
A103690
Triangle read by rows: T(n,k)=binomial(n,k-1)*k^(k-1)*(n+1-k)^(n-k) (1<=k<=n).
Original entry on oeis.org
1, 2, 4, 9, 12, 27, 64, 72, 108, 256, 625, 640, 810, 1280, 3125, 7776, 7500, 8640, 11520, 18750, 46656, 117649, 108864, 118125, 143360, 196875, 326592, 823543, 2097152, 1882384, 1959552, 2240000, 2800000, 3919104, 6588344, 16777216, 43046721
Offset: 1
- I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(p.27, Problem 1.2.6 (b)).
-
T:=proc(n,k) if k<=n then binomial(n,k-1)*k^(k-1)*(n+1-k)^(n-k) else 0 fi end: for n from 1 to 9 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
A206855
The sum of the degree of each root node over all rooted labeled trees on n nodes.
Original entry on oeis.org
0, 0, 2, 12, 96, 1000, 12960, 201684, 3670016, 76527504, 1800000000, 47158953820, 1362182012928, 43011849456888, 1474041721757696, 54493461914062500, 2161727821137838080, 91597537648314105376, 4128944057284204560384, 197293926880252878693804, 9961472000000000000000000
Offset: 0
-
nn=15;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];D[ Range[0,nn]!CoefficientList[Series[x Exp[y t],{x,0,nn}],x],y]/.y->1
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