A362857 -id:A362857 - cp's OEIS frontend

A362856 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-k)^(n-j) * j^j * binomial(n,j).

1, 1, 1, 1, 0, 4, 1, -1, 3, 27, 1, -2, 4, 17, 256, 1, -3, 7, 7, 169, 3125, 1, -4, 12, -9, 120, 2079, 46656, 1, -5, 19, -37, 121, 1373, 31261, 823543, 1, -6, 28, -83, 208, 797, 21028, 554483, 16777216, 1, -7, 39, -153, 441, 21, 14517, 373931, 11336753, 387420489

Offset: 0

Seiichi Manyama, May 05 2023

			Square array begins:
     1,    1,    1,   1,   1,     1, ...
     1,    0,   -1,  -2,  -3,    -4, ...
     4,    3,    4,   7,  12,    19, ...
    27,   17,    7,  -9, -37,   -83, ...
   256,  169,  120, 121, 208,   441, ...
  3125, 2079, 1373, 797,  21, -1525, ...

Eric Weisstein's World of Mathematics, Lambert W-Function.

Columns k=0..3 give A000312, (-1)^n * A069856(n), A362857, A362858.
Main diagonal gives A290158.
Cf. A362019.

T(n, k) = sum(j=0, n, (-k)^(n-j)*j^j*binomial(n,j));

E.g.f. of column k: exp(-k*x) / (1 + LambertW(-x)).

G.f. of column k: Sum_{j>=0} (j*x)^j / (1 + k*x)^(j+1).

A362859 Expansion of e.g.f. exp(-x) / (1 + LambertW(-2*x)).

Original entry on oeis.org

1, 1, 13, 173, 3321, 81529, 2443333, 86475493, 3529941873, 163260749681, 8437633695741, 481912844592541, 30142773978386281, 2049173019206244073, 150443505029536707381, 11862692305729094644949, 999864950902004743707873, 89709634016056661732903137

Offset: 0

Seiichi Manyama, May 05 2023

Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=2 of A362019.

Cf. A062971, A362857.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-x)/(1 + lambertw(-2*x))))

G.f.: Sum_{k>=0} (2*k*x)^k / (1 + x)^(k+1).

a(n) = (-1)^n * Sum_{k=0..n} (-2*k)^k * binomial(n,k).

cp's OEIS Frontend

A362856 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-k)^(n-j) * j^j * binomial(n,j).

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