A290158 -id:A290158 - 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).

A362836 Expansion of e.g.f. 1/(1 + LambertW(-x * (exp(x) - 1))).

Original entry on oeis.org

1, 0, 2, 3, 52, 245, 4086, 36547, 663832, 8984313, 184262770, 3334315391, 77900601780, 1751855308645, 46508427942718, 1241853335819475, 37195023972070576, 1144511291020453361, 38337497638919397738, 1331709923436162817447

Offset: 0

Seiichi Manyama, May 05 2023

nonn

Cf. A290158, A362835.

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

a(n) = n! * Sum_{k=0..floor(n/2)} k^k * Stirling2(n-k,k)/(n-k)!.

a(n) ~ n^n / (sqrt(exp(r)*(1+r) - 1) * r^(n + 1/2) * exp(n + 1/2)), where r = 0.528399250336668412340528181936966763473... is the root of the equation exp(1+r)-exp(1) = 1/r. - Vaclav Kotesovec, May 19 2025

A361652 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^j * Stirling2(n-j,j)/(n-j)!.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 6, 16, 0, 1, 0, 8, 9, 56, 65, 0, 1, 0, 10, 12, 120, 250, 336, 0, 1, 0, 12, 15, 208, 555, 1812, 1897, 0, 1, 0, 14, 18, 320, 980, 5148, 12614, 11824, 0, 1, 0, 16, 21, 456, 1525, 11064, 39711, 101040, 80145, 0

Offset: 0

Seiichi Manyama, May 05 2023

			Square array begins:
  1,  1,   1,   1,   1,    1, ...
  0,  0,   0,   0,   0,    0, ...
  0,  2,   4,   6,   8,   10, ...
  0,  3,   6,   9,  12,   15, ...
  0, 16,  56, 120, 208,  320, ...
  0, 65, 250, 555, 980, 1525, ...

Columns k=0..3 give: A000007, A052506, A351733, A351734.
Main diagonal gives (-1)^n * A290158(n).
Cf. A362834.

T(n, k) = n!*sum(j=0, n\2, k^j*stirling(n-j, j, 2)/(n-j)!);

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

A362862 a(n) = (-1)^n * Sum_{k=0..n} (-nk)^k binomial(n,k).

Original entry on oeis.org

1, 0, 13, 629, 58993, 8998399, 2035844461, 640881617123, 267995012680641, 143734541641235567, 96200314049944377901, 78599287990433271805699, 76993408916168689318057201, 89072357257840197226050646151

Offset: 0

Seiichi Manyama, May 06 2023

nonn

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

Main diagonal of A362019.

Cf. A290158.

Table[(-1)^n*(1 + Sum[(-n*k)^k*Binomial[n, k], {k, 1, n}]), {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)

a(n) = (-1)^n * sum(k=0, n, (-n*k)^k*binomial(n, k));

a(n) = n! * [x^n] exp(-x) / (1 + LambertW(-n*x)).

a(n) = [x^n] Sum_{k>=0} (n*k*x)^k / (1 + x)^(k+1).

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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A362836 Expansion of e.g.f. 1/(1 + LambertW(-x * (exp(x) - 1))).

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A361652 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^j * Stirling2(n-j,j)/(n-j)!.

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Formula

A362862 a(n) = (-1)^n * Sum_{k=0..n} (-nk)^k binomial(n,k).

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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).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A362836 Expansion of e.g.f. 1/(1 + LambertW(-x * (exp(x) - 1))).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A361652 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^j * Stirling2(n-j,j)/(n-j)!.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A362862 a(n) = (-1)^n * Sum_{k=0..n} (-n*k)^k * binomial(n,k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A362862 a(n) = (-1)^n * Sum_{k=0..n} (-nk)^k binomial(n,k).