cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, May 05 2023

Keywords

Examples

			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, ...

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Seiichi Manyama, May 05 2023

Keywords

Links

Crossrefs

Column k=2 of A362019.
Cf. A062971, A362857.

Programs

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

Formula

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).

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

Views

Author

Seiichi Manyama, May 05 2023

Keywords

Links

Crossrefs

Column k=3 of A362019.
Cf. A091482, A362858.

Programs

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

Formula

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).

A122087 Triangle read by rows: T(n,k) = number of unlabeled free bicolored trees with n nodes (n >= 1) and k (1 <= k <= floor(n/2), except k = 0 if n = 1 ) nodes of one color and n-k nodes of the other color (the colors are interchangeable).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 3, 1, 3, 7, 1, 3, 10, 9, 1, 4, 14, 28, 1, 4, 19, 45, 37, 1, 5, 24, 73, 132, 1, 5, 30, 105, 242, 168, 1, 6, 37, 152, 412, 693, 1, 6, 44, 204, 660, 1349, 895, 1, 7, 52, 274, 1008, 2472, 3927, 1, 7, 61, 351, 1479, 4219, 8105, 5097, 1, 8
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2006

Keywords

Examples

			K M N gives the number N of unlabeled free bicolored trees with K nodes of one color and M nodes of the other color.
0 1 1
Total( 1) = 1
1 1 1
Total( 2) = 1
1 2 1
Total( 3) = 1
1 3 1
2 2 1
Total( 4) = 2
1 4 1
2 3 2
Total( 5) = 3
1 5 1
2 4 2
3 3 3
Total( 6) = 6
1 6 1
2 5 3
3 4 7
Total( 7) = 11
1 7 1
2 6 3
3 5 10
4 4 9
Total( 8) = 23
From _Andrew Howroyd_, Apr 05 2023: (Start)
Triangle begins:
  n\k| 0 1  2   3    4    5    6
 ----+----------------------------
   1 | 1;
   2 | . 1;
   3 | . 1;
   4 | . 1, 1;
   5 | . 1, 2;
   6 | . 1, 2,  3;
   7 | . 1, 3,  7;
   8 | . 1, 3, 10,   9;
   9 | . 1, 4, 14,  28;
  10 | . 1, 4, 19,  45,  37;
  11 | . 1, 5, 24,  73, 132;
  12 | . 1, 5, 30, 105, 242, 168;
    ...
(End)

References

  • R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.

Links

Crossrefs

Row sums give A000055.
Cf. A119856, A329054, A362019 (labeled version).

Formula

T(n,k) = A329054(k, n-k) for 2*k < n; T(2*n,n) = A119856(n). - Andrew Howroyd, Apr 04 2023

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

Views

Author

Seiichi Manyama, May 06 2023

Keywords

Links

Crossrefs

Main diagonal of A362019.
Cf. A290158.

Programs

  • Mathematica
    Table[(-1)^n*(1 + Sum[(-n*k)^k*Binomial[n, k], {k, 1, n}]), {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)
  • PARI
    a(n) = (-1)^n * sum(k=0, n, (-n*k)^k*binomial(n, k));

Formula

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).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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