A362858 -id:A362858 - 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).

A362860 Expansion of e.g.f. exp(-x) / (1 + LambertW(-3*x)).

Original entry on oeis.org

1, 2, 31, 629, 18025, 662639, 29752957, 1578248867, 96577834801, 6696994946543, 518978239136341, 44448540938239811, 4169223860364566857, 425060509005908328071, 46801425208023247277965, 5534686715620432932442619, 699654866766940182167273185

Offset: 0

Seiichi Manyama, May 05 2023

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

Column k=3 of A362019.

Cf. A091482, A362858.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-x)/(1 + lambertw(-3*x))))

G.f.: Sum_{k>=0} (3*k*x)^k / (1 + x)^(k+1).

a(n) = (-1)^n * Sum_{k=0..n} (-3*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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