A351777 -id:A351777 - cp's OEIS frontend

A351762 Expansion of e.g.f. 1/(1 - 2xexp(x)).

1, 2, 12, 102, 1160, 16490, 281292, 5598110, 127326096, 3257961426, 92625793940, 2896747456262, 98827517418456, 3652643136982970, 145385563800940764, 6200097935648462190, 282035994269804870432, 13631368700936950044578, 697586352315912913754916

Offset: 0

Seiichi Manyama, Feb 18 2022

nonn

Column k=2 of A351761.

Cf. A000629, A351777.

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

a(n) = n!*sum(k=0, n, 2^(n-k)*(n-k)^k/k!);

a(n) = if(n==0, 1, 2*n*sum(k=0, n-1, binomial(n-1, k)*a(k)));

a(n) = n! * Sum_{k=0..n} 2^(n-k) * (n-k)^k/k!.
a(0) = 1 and a(n) = 2 * n * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.
a(n) ~ n! / ((1 + LambertW(1/2)) * LambertW(1/2)^n). - Vaclav Kotesovec, Feb 19 2022

A351776 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * (n-j)^j/j!.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 4, 3, 0, 1, -4, 12, -6, -4, 0, 1, -5, 24, -63, -8, -25, 0, 1, -6, 40, -204, 420, 150, 114, 0, 1, -7, 60, -465, 2288, -3435, -972, 287, 0, 1, -8, 84, -882, 7180, -32020, 33462, 3682, -4152, 0, 1, -9, 112, -1491, 17256, -138525, 537576, -379155, 6256, 1647, 0

Offset: 0

Seiichi Manyama, Feb 19 2022

			Square array begins:
  1,   1,   1,     1,      1,       1, ...
  0,  -1,  -2,    -3,     -4,      -5, ...
  0,   0,   4,    12,     24,      40, ...
  0,   3,  -6,   -63,   -204,    -465, ...
  0,  -4,  -8,   420,   2288,    7180, ...
  0, -25, 150, -3435, -32020, -138525, ...

Columns k=0..3 give A000007, A302397, A351777, A351778.
Main diagonal gives A351779.
Cf. A292861, A351761.

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

T(n, k) = if(n==0, 1, -k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));

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

T(0,k) = 1 and T(n,k) = -k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

cp's OEIS Frontend

A351762 Expansion of e.g.f. 1/(1 - 2xexp(x)).

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A351776 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * (n-j)^j/j!.

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cp's OEIS Frontend

A351762 Expansion of e.g.f. 1/(1 - 2*x*exp(x)).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

PARI

Formula

A351776 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * (n-j)^j/j!.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

PARI

Formula

A351762 Expansion of e.g.f. 1/(1 - 2xexp(x)).