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 1-6 of 6 results.

A382673 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] exp(x+y) / (exp(x) + exp(y) - exp(x+y))^3.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 22, 52, 22, 1, 1, 46, 208, 208, 46, 1, 1, 94, 736, 1372, 736, 94, 1, 1, 190, 2440, 7516, 7516, 2440, 190, 1, 1, 382, 7792, 37012, 60316, 37012, 7792, 382, 1, 1, 766, 24328, 170668, 418996, 418996, 170668, 24328, 766, 1, 1, 1534, 74896, 754132, 2653036, 3964684, 2653036, 754132, 74896, 1534, 1
Offset: 0

Views

Author

Seiichi Manyama, Apr 03 2025

Keywords

Examples

			Square array begins:
  1,  1,    1,     1,      1,       1, ...
  1,  4,   10,    22,     46,      94, ...
  1, 10,   52,   208,    736,    2440, ...
  1, 22,  208,  1372,   7516,   37012, ...
  1, 46,  736,  7516,  60316,  418996, ...
  1, 94, 2440, 37012, 418996, 3964684, ...
  ...

Crossrefs

Columns k=0..2 give A000012, A033484, A382675.
Main diagonal gives A382676.
Cf. A099594, A136126, A382674.
Cf. A382735.

Programs

  • PARI
    a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*stirling(n+1, j+1, 2)*stirling(k+1, j+1, 2));

Formula

E.g.f.: exp(x+y) / (exp(x) + exp(y) - exp(x+y))^3.
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+2,2) * Stirling2(n+1,j+1) * Stirling2(k+1,j+1).

A382734 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / (exp(x) + exp(y) - exp(x+y))^2.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 14, 2, 0, 0, 2, 38, 38, 2, 0, 0, 2, 86, 254, 86, 2, 0, 0, 2, 182, 1118, 1118, 182, 2, 0, 0, 2, 374, 4142, 8654, 4142, 374, 2, 0, 0, 2, 758, 14078, 51662, 51662, 14078, 758, 2, 0, 0, 2, 1526, 45614, 267566, 467102, 267566, 45614, 1526, 2, 0
Offset: 0

Views

Author

Seiichi Manyama, Apr 04 2025

Keywords

Examples

			Square array begins:
  1, 0,   0,    0,     0,      0, ...
  0, 2,   2,    2,     2,      2, ...
  0, 2,  14,   38,    86,    182, ...
  0, 2,  38,  254,  1118,   4142, ...
  0, 2,  86, 1118,  8654,  51662, ...
  0, 2, 182, 4142, 51662, 467102, ...

Crossrefs

Main diagonal gives A382737.
Cf. A371761, A382735, A382736.
Cf. A136126, A382740.

Programs

  • PARI
    a(n, k) = sum(j=0, min(n, k), j!*(j+1)!*stirling(n, j, 2)*stirling(k, j, 2));

Formula

E.g.f.: 1 / (exp(x) + exp(y) - exp(x+y))^2.
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} j! * (j+1)! * Stirling2(n,j) * Stirling2(k,j).

A382736 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / (exp(x) + exp(y) - exp(x+y))^4.

Original entry on oeis.org

1, 0, 0, 0, 4, 0, 0, 4, 4, 0, 0, 4, 44, 4, 0, 0, 4, 124, 124, 4, 0, 0, 4, 284, 1084, 284, 4, 0, 0, 4, 604, 5164, 5164, 604, 4, 0, 0, 4, 1244, 19804, 48044, 19804, 1244, 4, 0, 0, 4, 2524, 68524, 313804, 313804, 68524, 2524, 4, 0, 0, 4, 5084, 224284, 1707884, 3281404, 1707884, 224284, 5084, 4, 0
Offset: 0

Views

Author

Seiichi Manyama, Apr 04 2025

Keywords

Examples

			Square array begins:
  1, 0,   0,     0,      0,       0, ...
  0, 4,   4,     4,      4,       4, ...
  0, 4,  44,   124,    284,     604, ...
  0, 4, 124,  1084,   5164,   19804, ...
  0, 4, 284,  5164,  48044,  313804, ...
  0, 4, 604, 19804, 313804, 3281404, ...

Crossrefs

Main diagonal gives A382739.
Cf. A371761, A382734, A382735.
Cf. A382674, A382742.

Programs

  • PARI
    a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+3, 3)*stirling(n, j, 2)*stirling(k, j, 2));

Formula

E.g.f.: 1 / (exp(x) + exp(y) - exp(x+y))^4.
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+3,3) * Stirling2(n,j) * Stirling2(k,j).

A382738 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling2(n,k)^2.

Original entry on oeis.org

1, 3, 27, 579, 22779, 1396803, 121998267, 14333812419, 2175860165499, 414000255441603, 96422983358827707, 26970211126038920259, 8918364340126714711419, 3440770498298077165166403, 1531504734740033368269820347, 778873986278207207346380124099
Offset: 0

Views

Author

Seiichi Manyama, Apr 04 2025

Keywords

Crossrefs

Main diagonal of A382735.
Cf. A048144, A382737, A382739.
Cf. A382676.

Programs

  • Mathematica
    Table[Sum[k! * (k+2)! * StirlingS2[n,k]^2/2, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 30 2025 *)
  • PARI
    a(n) = sum(k=0, n, k!^2*binomial(k+2, 2)*stirling(n, k, 2)^2);

Formula

a(n) == 0 (mod 3) for n > 0.
a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x+y))^3.
a(n) ~ sqrt(Pi) * n^(2*n + 5/2) / (16 * sqrt(1 - log(2)) * exp(2*n) * log(2)^(2*n+3)). - Vaclav Kotesovec, Aug 30 2025

A382741 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/3) * (1 / (exp(x) + exp(y) - exp(x+y))^3 - 1).

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 25, 25, 1, 1, 57, 193, 57, 1, 1, 121, 889, 889, 121, 1, 1, 249, 3361, 7593, 3361, 249, 1, 1, 505, 11545, 47641, 47641, 11545, 505, 1, 1, 1017, 37633, 253737, 465601, 253737, 37633, 1017, 1, 1, 2041, 118969, 1228249, 3657721, 3657721, 1228249, 118969, 2041, 1
Offset: 1

Views

Author

Seiichi Manyama, Apr 04 2025

Keywords

Examples

			Square array begins:
  1,   1,     1,      1,       1,        1, ...
  1,   9,    25,     57,     121,      249, ...
  1,  25,   193,    889,    3361,    11545, ...
  1,  57,   889,   7593,   47641,   253737, ...
  1, 121,  3361,  47641,  465601,  3657721, ...
  1, 249, 11545, 253737, 3657721, 40666089, ...

Crossrefs

Cf. A272644, A382740, A382742.
Cf. A382735.

Programs

  • PARI
    a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*stirling(n, j, 2)*stirling(k, j, 2))/3;

Formula

E.g.f.: (1/3) * (1 / (exp(x) + exp(y) - exp(x+y))^3 - 1).
A(n,k) = A(k,n).
A(n,k) = (1/3) * A382735(n,k).

A382800 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / (1 - log(1-x) * log(1-y))^3.

Original entry on oeis.org

1, 0, 0, 0, 3, 0, 0, 3, 3, 0, 0, 6, 27, 6, 0, 0, 18, 78, 78, 18, 0, 0, 72, 282, 588, 282, 72, 0, 0, 360, 1272, 2988, 2988, 1272, 360, 0, 0, 2160, 6936, 16344, 24612, 16344, 6936, 2160, 0, 0, 15120, 44496, 101448, 175632, 175632, 101448, 44496, 15120, 0
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2025

Keywords

Examples

			Square array begins:
  1,  0,    0,     0,      0,       0, ...
  0,  3,    3,     6,     18,      72, ...
  0,  3,   27,    78,    282,    1272, ...
  0,  6,   78,   588,   2988,   16344, ...
  0, 18,  282,  2988,  24612,  175632, ...
  0, 72, 1272, 16344, 175632, 1669128, ...

Crossrefs

Main diagonal gives A382806.
Cf. A379821, A382799.
Cf. A382735, A382802.

Programs

  • PARI
    a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*abs(stirling(n, j, 1)*stirling(k, j, 1)));

Formula

E.g.f.: 1 / (1 - log(1-x) * log(1-y))^3.
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+2,2) * |Stirling1(n,j)| * |Stirling1(k,j)|.
Showing 1-6 of 6 results.

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

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