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.

A382735 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))^3.

Original entry on oeis.org

1, 0, 0, 0, 3, 0, 0, 3, 3, 0, 0, 3, 27, 3, 0, 0, 3, 75, 75, 3, 0, 0, 3, 171, 579, 171, 3, 0, 0, 3, 363, 2667, 2667, 363, 3, 0, 0, 3, 747, 10083, 22779, 10083, 747, 3, 0, 0, 3, 1515, 34635, 142923, 142923, 34635, 1515, 3, 0, 0, 3, 3051, 112899, 761211, 1396803, 761211, 112899, 3051, 3, 0
Offset: 0

Views

Author

Seiichi Manyama, Apr 04 2025

Keywords

Examples

			Square array begins:
  1, 0,   0,     0,      0,       0, ...
  0, 3,   3,     3,      3,       3, ...
  0, 3,  27,    75,    171,     363, ...
  0, 3,  75,   579,   2667,   10083, ...
  0, 3, 171,  2667,  22779,  142923, ...
  0, 3, 363, 10083, 142923, 1396803, ...

Crossrefs

Main diagonal gives A382738.
Cf. A371761, A382734, A382736.
Cf. A382673, A382741.

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));

Formula

E.g.f.: 1 / (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,j) * Stirling2(k,j).

A382737 a(n) = Sum_{k=0..n} k! * (k+1)! * Stirling2(n,k)^2.

Original entry on oeis.org

1, 2, 14, 254, 8654, 467102, 36414734, 3862847774, 534433092494, 93409669590302, 20117959360842254, 5233190283794276894, 1617259866279958581134, 585633786711715561283102, 245587300036701328750786574, 118067003149791582488105955614, 64502003996859329263691323378574
Offset: 0

Views

Author

Seiichi Manyama, Apr 04 2025

Keywords

Links

Crossrefs

Main diagonal of A382734.
Cf. A048144, A382738, A382739.

Programs

  • Maple
    f:= proc(n) local k;  add(k!*(k+1)!*Stirling2(n,k)^2, k=0..n) end proc:
map(f, [$0..40]);
  • Mathematica
    Table[Sum[k! * (k+1)! * StirlingS2[n,k]^2, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Apr 13 2025 *)
  • PARI
    a(n) = sum(k=0, n, k!*(k+1)!*stirling(n, k, 2)^2);

Formula

a(n) == 0 (mod 2) for n > 0.
a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x+y))^2.
a(n) ~ sqrt(Pi) * n^(2*n + 3/2) / (4 * sqrt(1 - log(2)) * exp(2*n) * log(2)^(2*n+2)). - Vaclav Kotesovec, Apr 13 2025

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).

A382740 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/2) * (1 / (exp(x) + exp(y) - exp(x+y))^2 - 1).

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 19, 19, 1, 1, 43, 127, 43, 1, 1, 91, 559, 559, 91, 1, 1, 187, 2071, 4327, 2071, 187, 1, 1, 379, 7039, 25831, 25831, 7039, 379, 1, 1, 763, 22807, 133783, 233551, 133783, 22807, 763, 1, 1, 1531, 71839, 636679, 1748791, 1748791, 636679, 71839, 1531, 1
Offset: 1

Views

Author

Seiichi Manyama, Apr 04 2025

Keywords

Examples

			Square array begins:
  1,   1,    1,      1,       1,        1, ...
  1,   7,   19,     43,      91,      187, ...
  1,  19,  127,    559,    2071,     7039, ...
  1,  43,  559,   4327,   25831,   133783, ...
  1,  91, 2071,  25831,  233551,  1748791, ...
  1, 187, 7039, 133783, 1748791, 18207367, ...

Crossrefs

Cf. A272644, A382741, A382742.
Cf. A382734.

Programs

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

Formula

E.g.f.: (1/2) * (1 / (exp(x) + exp(y) - exp(x+y))^2 - 1).
A(n,k) = A(k,n).
A(n,k) = (1/2) * A382734(n,k).

A382799 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))^2.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 4, 14, 4, 0, 0, 12, 40, 40, 12, 0, 0, 48, 144, 260, 144, 48, 0, 0, 240, 648, 1284, 1284, 648, 240, 0, 0, 1440, 3528, 6936, 9588, 6936, 3528, 1440, 0, 0, 10080, 22608, 42744, 65928, 65928, 42744, 22608, 10080, 0, 0, 80640, 166896, 300240, 476808, 581952, 476808, 300240, 166896, 80640, 0
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2025

Keywords

Examples

			Square array begins:
  1,  0,   0,    0,     0,      0, ...
  0,  2,   2,    4,    12,     48, ...
  0,  2,  14,   40,   144,    648, ...
  0,  4,  40,  260,  1284,   6936, ...
  0, 12, 144, 1284,  9588,  65928, ...
  0, 48, 648, 6936, 65928, 581952, ...

Crossrefs

Main diagonal gives A382804.
Cf. A379821, A382800.
Cf. A382734, A382801.

Programs

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

Formula

E.g.f.: 1 / (1 - log(1-x) * log(1-y))^2.
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} j! * (j+1)! * |Stirling1(n,j)| * |Stirling1(k,j)|.
Showing 1-5 of 5 results.

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