A347281
a(n) = 2^(n - 1)*permanent(M_n)^2 where M_n is the n X n matrix M_n(j, k) = cos(Pi*j*k/n).
Original entry on oeis.org
1, 2, 4, 0, 36, 288, 144, 18432, 11664, 115200, 144400, 0, 808151184, 133693952, 262440000, 299649466368, 7937314520976, 73575242956800, 21204146201616, 6459752448000000, 212406372892224, 8753824001424826368, 195844025123172289600, 152252829159294763008, 26487254903393025000000
Offset: 1
a(7) = -3384288*cos(Pi/7) - 3460896*sin(Pi/14) - 45888*cos(2*Pi/7) - 28224*cos(15*Pi/7) + 48384*cos(17*Pi/7) + 1706400 + 3458400*sin(3*Pi/14) = 144. - _Chai Wah Wu_, Sep 19 2021
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P(n)=matpermanent(matrix(n,n,j,k,cos((Pi*j*k)/n)));
for(k=1,25,print1(round(2^(k-1)*P(k)^2),", "))
A176158
Triangle read by rows: T(n,m) = (1 + 2 * binomial(n,m))^n for 0 <= m <= n, n >= 0.
Original entry on oeis.org
1, 3, 3, 9, 25, 9, 27, 343, 343, 27, 81, 6561, 28561, 6561, 81, 243, 161051, 4084101, 4084101, 161051, 243, 729, 4826809, 887503681, 4750104241, 887503681, 4826809, 729, 2187, 170859375, 271818611107, 9095120158391, 9095120158391, 271818611107, 170859375, 2187
Offset: 0
{1},
{3, 3},
{9, 25, 9},
{27, 343, 343, 27},
{81, 6561, 28561, 6561, 81},
{243, 161051, 4084101, 4084101, 161051, 243},
{729, 4826809, 887503681, 4750104241, 887503681, 4826809, 729},
{2187, 170859375, 271818611107, 9095120158391, 9095120158391, 271818611107, 170859375, 2187}.
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f:= proc(n) local m; seq((binomial(n,m)*2+1)^n, m=0..n) end proc:
for n from 0 to 10 do f(n) od; # Robert Israel, Dec 04 2024
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Clear[p, n, m];
p[x_, n_, m_] := (1 + 2*Binomial[n, m]*x)^n;
Table[Table[ Apply[Plus, CoefficientList[p[x, n, m], x]], {m, 0, n}], {n, 0, 10}];
Flatten[%]
A321968
a(n) = 2^n*n!*[x^n] -sqrt(exp(LambertW(-x)))*(LambertW(-x) + 1).
Original entry on oeis.org
-1, 3, 7, 55, 735, 13851, 336743, 10024911, 353109375, 14361853555, 662358958599, 34154042002983, 1947046027041503, 121593475341796875, 8255204941334951655, 605377094064557529151, 47687467918910168180223, 4015909348423983176411619
Offset: 0
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-sqrt(exp(LambertW(-x)))*(LambertW(-x) + 1): series(%, x, 32):
seq(2^n*n!*coeff(%, x, n), n=0..17);
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a[n_] := 2^n n! SeriesCoefficient[-Sqrt[Exp[ProductLog[-x]]] (ProductLog[ -x ] + 1), {x, 0, n}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 21 2019 *)
A364870
Array read by ascending antidiagonals: A(n, k) = (n + k)^n, with k >= 0.
Original entry on oeis.org
1, 1, 1, 4, 2, 1, 27, 9, 3, 1, 256, 64, 16, 4, 1, 3125, 625, 125, 25, 5, 1, 46656, 7776, 1296, 216, 36, 6, 1, 823543, 117649, 16807, 2401, 343, 49, 7, 1, 16777216, 2097152, 262144, 32768, 4096, 512, 64, 8, 1, 387420489, 43046721, 4782969, 531441, 59049, 6561, 729, 81, 9, 1
Offset: 0
The array begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
4, 9, 16, 25, 36, 49, ...
27, 64, 125, 216, 343, 512, ...
256, 625, 1296, 2401, 4096, 6561, ...
3125, 7776, 16807, 32768, 59049, 100000, ...
...
Cf.
A000012 (n=0),
A000169,
A000272,
A000312 (k=0),
A007830 (k=3),
A008785 (k=4),
A008786 (k=5),
A008787 (k=6),
A031973 (antidiagonal sums),
A052746 (2nd superdiagonal),
A052750,
A062971 (main diagonal),
A079901 (read by descending antidiagonals),
A085527 (1st superdiagonal),
A085528 (1st subdiagonal),
A085532,
A099753.
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A[n_,k_]:=(n+k)^n; Join[{1},Table[A[n-k,k],{n,9},{k,0,n}]]//Flatten
A376067
E.g.f. satisfies A(x) = (-log(1 - x / (1 - A(x))^2)) * (1 - A(x)).
Original entry on oeis.org
0, 1, 3, 26, 372, 7424, 190150, 5946576, 219643592, 9357076704, 451643892408, 24359462797680, 1451906224395792, 94769186402062080, 6723078079388867040, 515064037555614081024, 42380187502270667120640, 3727409807764337879016960
Offset: 0
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a(n) = sum(k=1, n, (2*n-2)!/(2*n-k-1)!*abs(stirling(n, k, 1)));
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